FIXED POINT FREE ACTIONS ON Z-ACYCLIC 2-COMPLEXES1 BOB OLIVER2 AND YOAV SEGEV3 In this paper, we give a complete description of the finite groups which can * *act on 2-dimensional Z-acyclic complexes without fixed points. One example of such an * *action (by the group A5) has been known for a long time, but as far as we know it is t* *he only such action constructed earlier. In fact, we construct here actions of this typ* *e for many different finite simple groups. More precisely, our main theorem is the following. Theorem A. For any finite group G, there is an essential fixed point free 2-d* *imensional (finite) Z-acyclic G-complex if and only if G is isomorphic to one of the simpl* *e groups P SL2(2k) for k 2, P SL2(q) for q 3 (mod 8) and q 5, or Sz(2k) for odd k 3. Furthermore, the isotropy subgroups of any such G-complex are all solvable. Here "G-complex" means a G-CW complex; but the same result holds if one inste* *ad uses simplicial complexes with admissible G-action in the sense of [S1] or [AS * *] (see Proposition A.4 in the appendix). The word "finite" is in parentheses because * *the theorem holds whether or not this condition is included. The condition that the* * action be essential was put in to insure that an action of a quotient group G=N does n* *ot automatically produce an action of G: Definition.A G-complex X is essential if there is no normal subgroup 1 6= N C G* * with the property that for each H G, the inclusion XHN ! XH induces an isomorphism on integral homology. In other words, if there is such a subgroup N C G, then the G-action on X is "essentially" the same as the G-action on XN , which factors through a G=N-acti* *on. In the case of actions on acyclic 2-complexes, the relation between essential acti* *ons and arbitrary actions is made precise in the next theorem. Theorem B. Let G be any finite group, and let X be any 2-dimensional Z-acycli* *c G- complex.0Let N be the subgroup generated by all normal subgroups N0 C G such th* *at XN 6= ;. Then XN is Z-acyclic; X is essential if and only if N = 1; and if N 6* *= 1 then the action of G=N on XN is essential. The proofs of Theorems A and B rely on the earlier works [O1 ], [O2 ], [S1], * *and [AS ], as well as on the classification theorem for finite simple groups. In [S1], Y. * *Segev proved that if a finite group G acts on an acyclic 2-complex X, the fixed point set XG* * is either Z-acyclic or empty, and is Z-acyclic if G is solvable or G ~=An for n 6. Later* *, in [AS ], ___________ 1Key words: fixed point free, 2-complex, finite simple group 1991 mathematics subject classification: primary 57S17, secondary 20D05, 57M20,* * 55M20 2Partially supported by the UMR 7539 of the CNRS. 3Partially supported by BSF grant no. 97-00042 and by grant no. 6782-1-95 f* *rom the Israeli Ministry of Science and Art. 1 2 Fixed point free actions on acyclic 2-complexes Aschbacher and Segev extended these results, and proved that XG 6= ; if G is si* *mple, except perhaps when G is of Lie type and Lie rank one, or the first Janko group* * J1 (a sporadic group). Techniques for constructing fixed point free actions of finite groups on fini* *te acyclic or contractible complexes (without restrictions on dimension) were developed by B.* * Oliver in several earlier papers such as [O1 ] and [O2 ]. In particular, in [O2 ], act* *ions for which the fixed point set of each subgroup is contractible or empty are studied. The proof of Theorem A _ both when constructing actions of G and when proving their nonexistence _ is based on refinements of the techniques developed in the* *se earlier papers of both authors. The main new input comes from a more detailed analysis * *of the subgroup lattice of G and its orbit space. In particular, necessary and suffici* *ent condi- tions for the existence of actions are stated in terms of this lattice in Propo* *sition 1.9. Afterwards, the proofs of nonexistence of actions of particular groups require * *identifying homology in certain "pieces" of the subgroup lattice of G. In fact, relatively few solvable subgroups need occur as isotropy groups for * *the actions constructed when proving Theorem A, and those which do occur are listed explici* *tly. It is possible that these and similarly constructed G-complexes can give new infor* *mation about decompositions of BG, and about the cohomology of G. Theorem A leaves open the question as to whether or not it is possible for a * *finite group to act on a 2-dimensional contractible complex without fixed points. Understan* *ding actions on acyclic 2-complexes is clearly a first step towards investigating th* *is question, but the first author feels that any serious attempt to answer it will require s* *ome very different methods than those used here. This paper is intended for both group theorists and topologists, and we have * *at- tempted to write it in a way which will be appealing and readable for both. In* * par- ticular, more background material has been included than might normally be the * *case, although we have tried to put most of that in the appendix at the end of the pa* *per. The paper is organized as follows. In Section 1, conditions are established, * *in terms of homological properties of the subgroup lattice of G, which determine the min* *imal dimensions of certain "universal" G-complexes. In particular, this section incl* *udes the general machinery for constructing such actions. After proving some technical r* *esults in Section 2, the constructions of the G-complexes described in Theorem A are carr* *ied out in Section 3. In Section 4, we show that any finite group G which acts essentia* *lly on a 2-dimensional acyclic complex must be almost simple (i.e., there is a nonabelia* *n simple group L such that L G Aut(L)). In Section 5, we develop machinery to show the nonexistence of actions on acyclic 2-complexes; and this is applied in Section * *6 to prove Theorem A for simple groups of Lie type and Lie rank one. The sporadic groups * *are dealt with in Theorem 7; except for the first Janko group J1 this repeats resul* *ts already shown in [AS ]. Theorem B is proven in Section 4, and Theorem A in Section 8. * * All of this is preceded by a preliminary "Section 0" where we present some general * *results about G-posets and construction of G-complexes; and is followed by an appendix * *which includes background material about G-complexes, Z[G]-modules, and simple groups* * of Lie type, as well as a sketch of the proofs in [S1] and [AS ] of certain cases * *of Theorem A. References of the form A.x, B.x, etc. all refer to the appendix. After the a* *ppendix, we attach a list of the notation used throughout the paper. Bob Oliver and Yoav Segev * * 3 Bob Oliver would like to thank the Hebrew University for the hospitality duri* *ng his visit in 1997. This visit, and the visit of Yoav Segev to Arhus University in 1* *992, played an important role in starting our collaboration. We would also like to thank Je* *an-Pierre Serre for his letters, which also helped revive our interest in this problem. 0. G-complexes and G-posets Posets, and in particular families of subgroups considered as posets, will pl* *ay an important role as "bookkeeping" devices for controlling dimensions of certain a* *cyclic complexes. For any poset S, we let N (S) denote its nerve: the simplicial compl* *ex with one vertex for each element of S, and one n-simplex for each chain ff0 < ff1 < * *. .<.ffn of elements of S. By a G-poset is meant a poset with G-action which preserves * *the ordering. A terminal subposet of a poset S is a subset S0 S such that fi ff * *2 S0 implies fi 2 S0. For any element ff in a poset S, we set Sff = {fi 2 S | fi ff* *}. The next lemma provides a general setting for comparing G-complexes with coverings * *to the nerves of the coverings. Lemma 0.1. Let X be a G-complex, let S be a finite G-poset, and let {Xff}ff2S* *be a covering of X by subcomplexes which satisfies the following conditions: (a) ff fi implies Xff Xfi. (b) For all x 2 X, the set {ff 2 S | x 2 Xff} has a largest element. (c) Xg(ff)= g(Xff) for all ff 2 S, g 2 G. Then there is a G-map fX : X ! N (S) with the property that fX (Xff) N (Sff) for all ff 2 S. (1) If, furthermore, Xffis acyclic (contractible) for each ff, then for any map f :* * X ! N (S) which satisfies (1), and any terminal subposetSS0 S, f restricts to a hom* *ology equivalence (homotopy equivalence) fS0: XS0= ff2S0Xff! N (S0). Proof.For each n 0, let Jn denote the G-set of n-cells of X, and let 'n: Jnx D* *n ! X denote the characteristic map for the n-cells. Let : Jn ! S be the map which s* *ends j 2 Jn to the largest element in the set {ff 2 S | 'n(j; 0) 2 Xff}; this is wel* *l defined by (b) and equivariant by (c). For each ff 2 S, we let [ff] denote the correspondi* *ng vertex in N (S). First define f0: X(0)! N (S) by setting f0('0(j; 0)) = [(j)] for each j 2 J0.* * This clearly satisfies condition (1). Now assume that fn-1: X(n-1)! N (S) has been defined, satisfying (1). For any j 2 Jn and any v 2 Sn-1, 'n(j; 0) 2 X(j) by construction, and so 'n(j; v) 2 X(j* *) since X(j) is a subcomplex of X. So fn-1('n(j; v)) 2 N (S(j) ) by (1), hence it is in* * some simplex which contains the vertex [(j)], and the segment from fn-1('n(j; v)) to* * [(j)] lies in N (S). So we can define fn: X(n)-----! N (S) 4 Fixed point free actions on acyclic 2-complexes by setting fn(x) = fn-1(x) for x 2 X(n-1), and fn('n(j; tv)) = t.fn-1('n(j; v)) + (1 - t).[(j)] for j 2 Jn, v 2 Sn-1, t * *2 [0;:1] This is well defined as a map of sets, since the two definitions agree on 'n(Jn* *x Sn-1) X(n-1). So it is continuous by Lemma A.3 (fn|X(n-1)and fn O'n are both continuo* *us). Condition (1) still holds for fn, since for all j 2 Jn and v 2 int(Dn), and all* * ff 2 S, 'n(j; v) 2 Xff () 'n(j; 0) 2 Xff=) ff (j) =) fn('n(j; v)) 2 N (S(j) ) N (Sff): And fn is equivariant since is equivariant, since fn-1 is equivariant (by indu* *ction), and since the G-action on N (S) is affine. Finally, define fX : X ! N (S) to be the union of the fn; this is again conti* *nuous by Lemma A.3, and condition (1) holds since it holds for each fn. Now let f be any map which satisfies (1), and assume that Xffis acyclic (cont* *ractible) for each ff 2 S. We want to show that f is a homology (homotopy) equivalence. The group action no longer plays a role here, so we can assume G = 1. We can assume inductivelySthat for any properly contained terminal poset S0 $ S, f res* *tricts to an equivalence ff2S0Xff! N (S0) (since the subspace and subposet still sat* *isfy conditions (a) and (b) above). If S has a smallest element oe, then X = Xoeis a* *cyclic (contractible) and N (S) is contractible, so any map f : X ! N (S) is a homolo* *gy (homotopy) equivalence, and we are done. Assume now that S contains no smallest element. In this case, we can write S* * = S1 [ S2, whereSS1 and S2 are proper terminal subposets of S. Set S0 = S1 \ S2; and set Xi = ff2SiXfffor each i = 0; 1; 2. Clearly, N (S0) = N (S1) \ N (S2)* *, and condition (b) implies that X0 = X1 \ X2. By the inductive assumption, f restric* *ts to homology (homotopy) equivalences fi: Xi! N (Si), and so f is a homology (homoto* *py)_ equivalence by Proposition B.3. |* *__| By a family of subgroups of G will here be meant any subset F S(G) which is closed under conjugation. We do not assume here that subgroups of elements of * *the family are also in the family. For any family F of subgroups of G, a (G; F)-complex will mean a G-CW-complex all of whose isotropy subgroups lie in F. A (G; F)-complex is universal if the* * fixed point set of each subgroup in F is contractible. (The "universality" property * *of such spaces is explained in Proposition A.6.) One can, in fact, construct universal * *(G; F)- complexes for any family F of subgroups of G, but in most cases any such comple* *x must be infinite dimensional. For example, when F = {1} contains only the trivial su* *bgroup, a universal (G; F)-complex is just a contractible complex upon which G acts fre* *ely; and so its orbit space is a classifying space for G. The results in Section 1 will * *make it clear what conditions are needed on F for there to be a finite (or finite dimensional* *) universal (G; F)-complex. The following lemma is the starting point for the constructions of universal * *(G; F)- complexes, and of other G-complexes satisfying certain homological conditions. * *Roughly, it describes the effect on the homology of X of attaching cells of one orbit ty* *pe G=H to X. By "attaching an orbit of cells of type G=Hx Di" to a G-complex X, we mean Bob Oliver and Yoav Segev * * 5 replacing X by the complex X ['(G=Hx Dn) for some G-map ': G=Hx Sn-1 ! X(n-1). We refer to Lemma A.2 for more detail. Proposition 0.2. Fix a finite G-complex X, and a subgroup H G. Then the follow- ing hold. (a) For any n 1, there is a finite G-complex Y X, obtained by attaching to X orbits of cells of type G=Hx Di for 1 i n, such that Y H is (n - 1)-connected* * and Hi(Y H) ~=Hi(XH ) for all i > n. Also, Hn(Y H) is Z-free if Hn(XH ) is Z-free. (b) Assume n 1, and that XH is (n - 1)-connected. For any homomorphism ': (Z[N(H)=H])k -----! Hn(XH ) of Z[N(H)=H]-modules, there is a finite G-complex Y X, obtained by attaching k orbits of cells G=Hx Dn+1 to X, such that Hi(Y H) ~=Hi(XH ) for all i 6= n; n +* * 1, such that Hn(Y H) ~=Coker('); (1) and such that there is a short exact sequence 0 ---! Hn+1(XH ) ----! Hn+1(Y H) ----! Ker (') ---! 0: (2) (c) Assume, for some n 1, that eH*(XH ) = Hn(XH ) is a stably free Z[N(H)=H]- module; more precisely that Hn(XH ) (Z[N(H)=H])k ~=(Z[N(H)=H])m (where k; m 0). Then there exists a G-complex Y X, obtained by attaching to X k orbits of cells of type G=Hx Dn and m orbits of cells of type G=Hx Dn+1, such* * that Y H is acyclic. (d) Assume that all connected components of XH are acyclic, and that one of * *the components of XH is fixed by the action of N(H)=H and the others are permuted f* *reely. Then there exists a G-complex Y X, obtained by attaching to X cells of orbit t* *ype G=Hx D1, such that Y H is acyclic. Proof. (b) Since XH is (n - 1)-connected, the Hurewicz theorem applies to show * *that each element h 2 Hn(XH ) is represented by a map ': Sn ! XH , in the sense that h = '*([Sn]) for some fixed generator [Sn] of Hn(Sn). (See, e.g., [Hu , Theorem* * II.9.1] if n > 1, or [Hu , Theorem II.6.1] if n = 1). And we can assume that '(Sn) (XH* * )(n) by the cellular approximation theorem [LW , Theorem II.8.5], which says that an* *y map Sn ! XH is homotopic to a cellular map, and in particular a map with image in * *the n-skeleton. Now let E = {e1; : :;:ek} denote the canonical basis of (Z[N(H)=H])k, and fix* * maps fi: Sn ! (XH )(n)which represent '(ei) 2 Hn(XH ). Define f : (Ex G=H)x Sn -----! X(n) by setting f(ei; gH; x) = g.fi(x); and let fH be the restriction of f to the H* *-fixed point sets. In particular, for each i and each g 2 N(H), f|eixgHxSn(as a map Sn* * ! XH ) represents the class g.'(ei) 2 Hn(XH ). In other words, Hn(fH ) = ' under* * the identification Hn((Ex G=H)H xSn) = Hn((Ex N(H)=H)x Sn) ~=(Z[N(H)=H])k: 6 Fixed point free actions on acyclic 2-complexes Set Y = X [f (Ex G=H)x Dn+1 (Lemma A.2). Then n+1 Y H = XH [fH (Ex N(H)=H)x D ; and (1) and (2) now follow from Lemma B.1. (a) We prove this inductively. Fix n 0 such that XH is (n - 1)-connected. * * We will construct a finite G-complex Y X, obtained by attaching orbits of cells o* *f type G=Hx Dn+1 to X, such that Y H is n-connected. If n = 0 and XH is not connected, then let v-1 and v1 be two vertices in dif* *ferent connected components of XH , define f : G=Hx S0 ! X by setting f(gH; t) = gvt, * *and set X0 = X [f (G=Hx D1). By construction, (X0)H has fewer connected components than XH , and by continuing the procedure we obtain a finite G-complex Y such t* *hat Y H is connected. If n = 1 and ss1(XH ) 6= 1, then choose any element 1 6= OE 2 ss1(XH ), repre* *sent it by a map f0: S1 ! XH , and extend this to a G-map f : G=Hx S1 ! X by setting f(gH; v) = g.f0(v). Set X0 = X [f (G=Hx D2). Then ss1((X0)H ) = ss1(XH )=N, whe* *re N is a normal subgroup of ss1(XN ) which contains OE (in fact, the normal closu* *re of ). Since ss1(XH ) is finitely generated, we can repeat this procedure and obtain a* * finite G-complex Y such that Y H is 1-connected. If n > 1, then the result follows from part (b), where we choose ' to be any * *surjection (Hn(XH ) is finitely generated as an abelian group, hence as a Z[N(H)=H]-module* *). (c) Upon applying point (b) to the trivial homomorphism '0: (Z[N(H)=H])k ! Hn-1(XH ) = 0; we get a finite G-complex Y0 X, obtained by attaching k-orbits of cells G=Hx D* *n to X, such that Hi((Y0)H ) ~=Hi(XH ) = 0 for all i 6= n and Hn((Y0)H ) ~=Hn(XH ) (Z[N(H)=H])k ~=(Z[N(H)=H])m : If we now apply (b) to any isomorphism ': (Z[N(H)=H])m ! Hn((Y0)H ), we obtain a finite G-complex Y Y0, constructed by attaching m orbits of cells G=Hx Dn+1, s* *uch that Y H is acyclic. (d) Here, we assume that all connected components of XH are acyclic, and that* * one is invariant under the action of N(H)=H and the others are permuted freely. Let X0* * XH denote the component which is N(H)=H-invariant, and let X1; X2; : :;:Xk be N(H)* *=H- orbit representatives for the other components. (If N(H)=H = 1, then let X0 be * *any of the connected components.) Fix vertices xi 2 Xi for i = 0; : :;:k. Set J = {1; * *: :;:k}, and define ': (G=Hx J)x S0 ! X by setting '(gH; i; 1) = gxi and '(gH; i; -1) = gx0: Now set Y = X [' ((G=Hx J)x D1). Then Y H = XH ['|((N(H)=Hx J)x D1); and this is acyclic since X0 has been connected (by a unique 1-cell) to each of* *_the other connected components of X. |__| Bob Oliver and Yoav Segev * * 7 We finish the section with two lemmas which involve elementary properties of * *nerves of posets. We first recall the following results of Quillen. Lemma 0.3 [Q2 , 1.3-1.5].(a) Let T S be posets, and let r : S ! T be any order preserving map such that r|T = IdT, and such that r(ff) ff for all ff 2 S (or * *r(ff) ff for all ff). Then the inclusion of N (T ) in N (S) is a homotopy equivalence. (b) Let G be a finite group, and let H be any set of subgroups of G. Assume t* *here is some H0 2 H such that either H \ H0 2 H for all H 2 H, or 2 H for all H 2 H. Then N (H) is contractible. Proof.Point (a) is shown in [Q2 , 1.3]. In fact, N (T ) is a strong deformation* * retract of N (S), where r induces the retraction N (S) ! N (T ), and where the homotopy wi* *th the identity comes from the assumption that r(x) is always x or always x [Q* *2 , 1.3]. If H is as in (b), then its nerve is "conically contractible" in the sense of* * Quillen,_and hence is contractible [Q2 , 1.4-1.5]. * * |__| The following lemma will also be useful, when showing that certain subgroups * *of G need not occur as isotropy subgroups in acyclic G-complexes. Lemma 0.4. Let S be any finite poset, and let S0 S be any subposet with the property that N (S>ff) ' * for all ff 2 Sr S0. Then N (S0) ' N (S) (the inclus* *ion induces a homotopy equivalence). Proof.It suffices to show this when Sr S0 contains just one element ff. In thi* *s case, N (S) is the union of N (S0) with the cone over the subcomplex A N (S0), where A = N Sff = N (Sff): (1) * Note that the nerve of the disjoint union in (1) is identified with the join of* * the nerves, since every element in Sff. Then A is contra* *ctible,_ since N (S>ff) ' * by assumption. * *|__| Lemma 0.4 does, in fact, hold without the assumption that S is finite: it fol* *lows as a consequence of Quillen's Theorem A [Q1 ] (see also [Q2 , Proposition 1.6]). A central problem throughout this paper, especially in Sections 5 and 6, is t* *o find ways to detect 2-dimensional homology in nerves of certain posets. Given a 2-cy* *cle in N (S), the simplest way to show it is nonvanishing in H2(N (S)) is to show that* * some 2-simplex with nonzero coefficient is maximal in N (S); i.e., not in the bounda* *ry of any 3-simplex. The following lemma provides a refinement of this observation, and w* *ill be used in Section 5. Lemma 0.5. Let S be a finite poset, and let z be a 2-cycle in the nerve of S.* * Fix ele- ments m < M in S, where m is minimal and M is maximal. Set Q = {x2 S | m< x< M}, and let Q0 Q be the set of all x 2 Q such that the simplex (m; x; M) occurs wi* *th nonzero coefficient in z. Assume that Q0 6= ;, and that some element of Q0 lie* *s in a separate connected component of N (Q) from all of the other elements of Q0. Th* *en 0 6= [z] 2 H2(N (S)). Proof.Set X = N (S), for short, and let Y X be the subcomplex of all simplices which do not contain both vertices m; M. Let C*(X) C*(Y ) be the simplicial ch* *ain 8 Fixed point free actions on acyclic 2-complexes complexes; and write X z = ax(m; x; M) (mod C2(Y )) x2Q0 (where 0 6= ax 2 Z for each x). For any 3-simplex oe in X, either oe is in Y (and so @(oe) 2 C2(Y )), or oe =* * (m; x; y; M) for some x; y 2 Q in the same connected component of N (Q) and @(oe) = (m; x; M) - (m; y; M) (mod C2(Y )): Thus, if z is a boundary, then the sum of the coefficients ax in the above expr* *ession for z, taken over all x 2 Q0 which lie in any given connected component of N (Q* *), is zero. And this contradicts the assumption that some element of Q0is in a compon* *ent_ by itself. |* *__| 1.Minimal dimensions of universal G-spaces We will now establish necessary and sufficient conditions for the existence o* *f universal complexes satisfying certain dimensional restrictions. These conditions will be* * expressed in terms of the homology of the nerves of certain posets. Throughout this section, G will be a finite group. A nonempty family F S(G) will be called separating if it has the following three properties: (a) G =2F; * *(b) any subgroup of an element of F is in F; and (c) for any H C K G with K=H solvable, K 2 F if H 2 F. The following property of separating families is immediate. Lemma 1.1. Each maximal subgroup in a separating family of subgroups of G is_* *self- normalizing. |__| If G is solvable, then it has no separating family of subgroups. If G is not * *solvable, then we let SLV denote the family of solvable subgroups: the minimal separatin* *g family for G. We also let MAX denote the maximal separating family for G, which can* * be described as follows. Let L be the maximal normal perfect subgroup of G; i.e., * *the last term in the derived series of G. Then MAX is the family of all subgroups of * *G which do not contain L. In particular, if G is perfect, then MAX is the family of * *all proper subgroups of G. A (G; F)-complex will be called H-universal if the fixed point set of each H * *2 F is acyclic. The importance of universal, and H-universal, (G; F)-complexes when st* *udying 2-dimensional actions comes from the following lemma. Lemma 1.2. Let X be any 2-dimensional acyclic G-complex without fixed points.* * Let F be the set of subgroups H G such that XH 6= ;. Then F is a separating family* * of subgroups of G, and X is an H-universal (G; F)-complex. Proof.By [S1, Theorem 3.4], XH is acyclic for each H 2 F; i.e., for each H suc* *h that XH 6= ;. (Another proof of this, which does not depend on the odd order theorem* *, is given in Theorem 4.1 here.) So by definition, X is an H-universal (G; F)-comple* *x. Also, if H C K G are subgroups such that H 2 F and K/H is solvable, then XH is acycl* *ic, Bob Oliver and Yoav Segev * * 9 and so XK = (XH )K=H is acyclic by [S1, Theorem 3.1] (see also Theorem 4.1). T* *hus,_ F is a separating family. * *|__| For any family F of subgroups of G, we consider N (F) as a G-complex via the conjugation action. Note, however, that N (F) is not itself a (G; F)-complex in* * general. For example, when F = {1}, then N (F) is a point, while a (G; F)-complex must h* *ave a free G-action. Recall that for any family F of subgroups of G and any set H of subgroups, FH denotes the poset of those subgroups in F which contain some element of H. Als* *o, for any H G, FH and F>H denote the posets of subgroups in F which contain H, or strictly contain H, respectively. The following proposition follows immediat* *ely from Lemma 0.1. Proposition 1.3. Fix any family F of subgroups of G. Let N (F) be the nerve of * *the poset F, regarded as a G-complex via the action by conjugation. Then for any (G* *; F)- complex X, there is a G-map f : X ! N (F) with the property that f(XH ) N (FH * * ) for all H G. And if X is universal (H-universal), then for any set H of subgro* *ups of G, any such map f restricts to a homotopy equivalence (homology equivalence) XH* * ! N (FH ). Proof.We apply Lemma 0.1, with S = F (regarded as a poset via inclusion), and XH = XH for H 2 F. Since X is a (G; F)-complex, every element of X is fixed by some H 2 F, and so {XH }X2F is a covering of X. Condition (a) of Lemma 0.1 clea* *rly holds, and condition (b) holds since the largest element of {H 2 F | x 2 XH } i* *s the -1 H * * __ isotropy group Gx. And condition (c) holds since XgHg = g(X ). * * |__| The following lemma, which helps to limit the number of orbit types needed wh* *en constructing "minimal" universal (G; F)-complexes, is an easy consequence of Le* *mma 0.4. Lemma 1.4. Let F be any family of subgroups of G, and let F0 F be any subfam* *ily such that N (F>H ) ' * for all H 2 Fr F0. Then any (H-)universal (G; F0)-comple* *x is also an (H-)universal (G; F)-complex; and N ((F0)H ) ' N (FH ) (1) for any set H of subgroups of G. Proof. For any set H of subgroups of G, point (1) follows from Lemma 0.4, appli* *ed to the posets S def=FH and S0def=(F0)H . Let X be an (H-) universal (G; F0)-complex. All isotropy subgroups of X lie * *in F0 F, so X is also a (G; F)-complex. For each K 2 F, XK is homotopy (homology) equivalent to N ((F0)K ) by Proposition 1.3 (applied with H = {K}); this in tu* *rn is homotopy (homology) equivalent to N (FK ) by (1); and this last complex is con* *tractible_ (acyclic). So X is also (H-) universal as a (G; F)-complex. * * |__| We are now ready to deal directly with the problem of controlling the dimensi* *ons of universal or H-universal (G; F)-complexes. This will be done by attaching cell* *s, one orbit type at a time, at each stage arranging for the appropriate fixed point s* *et to be contractible or acyclic. The key problem is how to do this with cells in free * *orbits. 10 Fixed point free actions on acyclic 2-complexes This will be described in the following three lemmas. The first will be needed* * when constructing contractible 1-complexes. Lemma 1.5. Let X be any finite G-set with the property that |XH | = 1 for each subgroup 1 6= H G of prime power order. Then X has one fixed point and is otherwise free. Proof. We may assume that XG = ;; otherwise the result is clear. We may also as* *sume that X has no free orbits (otherwise just remove them). By assumption, each Sy* *low subgroup of G acts freely on X away from one fixed point; and so |X| 1 (mod |G* *|). Write X = G=H1 q G=H2 q . .q.G=Hk, where 1 6= Hi$ G for all i. In particular, Xk [G:Hi] = |X| = r.|G| + 1 (1) i=1 for some r. Furthermore, for each pair of distinct elements x; y 2 X, the isot* *ropy subgroups Gx and Gy have trivial intersection, since otherwise Gx \ Gy contains* * a nontrivial p-subgroup (some p) which fixes two points of X. It follows that X Xk Xk |G| - 1 (|Gx| - 1) = [G:Hi].(|Hi| - 1) = k.|G| - [G:Hi]:(2) x2X i=1 i=1 Upon adding (1) and (2), we see that (2) is an equality, and that r = k - 1. Bu* *t then after dividing (1) by |G|, we get that Xk _1__ > k - 1: i=1|Hi| * * __ Since |Hi| 2 for all i, we must have k = 1, and hence |X| = 1. * * |__| A complex X will be called homologically m-dimensional if Hn(X) = 0 for all n* * > m, and Hm (X) is Z-free. (Technically, this should be called homologically m-dime* *nsional, since it only provides an upper bound on the degrees of homology of X.) We note* * first the following properties of subcomplexes of acyclic complexes. Lemma 1.6. Let X be any m-dimensional acyclic CW complex (m 1). Then any subcomplex of X is homologically (m- 1)-dimensional. And if A1; : :;:An X are homologically (m- 2)-dimensional subcomplexes, then their intersection is also * *homo- logically (m- 2)-dimensional. Proof.For any subcomplex A X, eHi(A) ~=Hi+1(X; A) must be zero for i m and Z-free for i = m - 1. Hence A is homologically (m- 1)-dimensional. It suffices to prove the last statement when n = 2. For each i m - 2, there* * is a Mayer-Vietoris exact sequence 0 ---! Hi+1(A1 [ A2) ----! Hi(A1 \ A2) ----! Hi(A1) Hi(A2): If i m - 1, then the first and last groups are zero, and so Hi(A1 \ A2) = 0. * *And if i = m - 2, then the first and last groups are Z-free, and so Hm-2(A1 \ A2)_i* *s_also Z-free. |__| The next lemma is essentially included in the proof of [O2 , Proposition 6]. Bob Oliver and Yoav Segev * * 11 Proposition 1.7. Let X be a finite G-complex with the following two properties. (a) For each 1 6= H G, XH is acyclic or empty, and is acyclic if H has prime* * power order. (b) For some n > 0, eH*(X) = Hn(X), and is Z-free. Then Hn(X) is stably free as a Z[G]-module. Proof. For each prime p and each Sylow p-subgroup S G, consider the subcomplex [ X0= XH = {x 2 X | Sx 6= 1}: 16=HS By Proposition 1.3, applied with H = {1 6= H S}, X0 is acyclic (N (H) ' * sinc* *e H has maximal element S). Hence H*(X; X0) ~=He*(X) also vanishes in degrees diffe* *rent from n. Furthermore, since all cells in Xr X0 are permuted freely by S, C*(X; X* *0) is a chain complex of finitely generated free Z[S]-modules (Lemma C.1). So by Propos* *ition C.2, the unique nonvanishing homology group Hn(X; X0) ~=Hn(X) is Z[S]-stably fr* *ee. (Since all but one summand in (1) of Proposition C.2 is stably free, so is the * *remaining summand, by definition.) In particular, Hn(X) is a Z[G]-module which is projec* *tive after restriction to each Sylow subgroup, and is hence Z[G]-projective by Rim's* * theorem [Rim , Proposition 4.9]. Now set Y = Xx X, where X is the unreduced suspension of X (see Lemma A.5). We identify X with the subcomplex Xx {x0} of Y , where x0 2 X is one of the suspension vertices. Then H*(X; x0) = Hn+1(X; x0) ~= Hn(X); and so by the K"unneth formula 8 >: 0 otherwise. Consider the subcomplexes [ [ Xs = XH and Ys = Y H: 16=HG 16=HG We claim that the inclusion map Xs ,! Ys is a homology equivalence. To see this* *, set F = {1 6= H G | XH 6= ;}. By Proposition 1.3, there is a map f : Ys ! N (F) such that f((Ys)H ) N ((F)H ) for all H G; and f|Xs has the same property. S* *ince Xs and Ys are both H-universal (G; F)-complexes (Y H = XH xXH is acyclic if XH is), Proposition 1.3 implies that f restricts to homology equivalences Ys ! N (* *F) and Xs ! N (F); and thus that the inclusion Xs Ys is a homology equivalence. In particular, this shows that H*(Y; X) ~= H*(Y; X [ Ys) (see Lemma B.2). Th* *us, C*(Y; X [ Ys) is a chain complex of free Z[G]-modules (by Lemma C.1, since G ac* *ts freely on Y r(X [ Ys)) with only two nonzero homology groups. Since Hn(X) Z Hn(* *X) is stably free by Proposition C.3, the other homology group Hn(X) must also be * *stably_ free by Proposition C.2. |* *__| For any G-space X and any H G, we write X>H = {x 2 X | Gx % H}: 12 Fixed point free actions on acyclic 2-complexes i.e., the union of fixed point sets of subgroups which strictly contain H. Also* *, for any family F S(G), F>H denotes the set of elements of F which strictly contain H. Proposition 1.8. Let G be any finite group, and let F be a separating family fo* *r G. Let F0 F be any subfamily with the property that N (F>H ) is contractible (and non* *empty) for all H 2 Fr F0. Let d: F0 ! N be any function which is constant on conjugacy classes of subgroups, such that d(H) = 0 for H maximal in F, such that N ((F0)>* *H ) is homologically (d(H)- 1)-dimensional for each non-maximal subgroup H 2 F0, and such that d(H) d(H0) whenever H H0. Then there is a finite H-universal (G; F0* *)- complex X with the property that dim(XH ) d(H) for each H 2 F0. Furthermore, X can be taken to be universal if d(H) 6= 2 for each H 2 F0. Also, X can be chose* *n such that every vertex of X is fixed by some maximal subgroup in F. Proof.Let Fmax be the set of maximal subgroups in F. Set X0 = Fmax, regarded as a zero-dimensional G-complex. Since the elements of Fmax are all self-norma* *lizing (Lemma 1.1), this is a 0-dimensional (G; F)-complex, and (X0)H contains exactly* * one point for each H 2 Fmax. Let H1; : :;:Hk = 1 be conjugacy class representative* *s for the elements of Fr Fmax, ordered such that d(H1) d(H2) . .d.(Hk), and such th* *at i j if Hi contains a subgroup conjugate to Hj. For each i = 0; : :;:k, let Hi * *be the set of all maximal subgroups in F, together with all subgroups conjugate to Hj * *for any j i. In particular, H0 = Fmax and Hk = F. We construct a sequence of G-complex* *es X0 X1 X2 . . .Xk, such that for each i 1, (a) dim(Xi) d(Hi) and X(0)i= X(0)0, (b) XirXi-1has only orbit types G=Hi, (c) Xi= Xi-1if Hi2=F0, and (d) (Xi)Hi is acyclic, and is contractible if Hi2 F0 and d(Hi) 6= 2. Note that for each H 2 Fmax, (X0)H = {H} is contractible, and hence (Xi)H will * *be contractible for all i > 0. Once the Xi have been constructed, we set X = Xk. * * This is a (G; F0)-complex; and for all H 2 F0, dim(XH ) d(H), and XH is acyclic, a* *nd contractible if d(H) 6= 2. And by (a), each vertex of X is in X0, and hence fix* *ed by a maximal subgroup of F. It remains to construct the Xi. Assume that Xi-1has been constructed (i 1). * *Then Xi-1is an H-universal (G; Hi-1)-complex. By Proposition 1.3 (and by definition * *of the Hj), H >H H* (Xi-1) i = H* (Xi-1) i ~=H* N ((Hi-1)>Hi) = H* N (F>Hi) : In particular, by Lemma 1.4, (Xi-1)Hi is homologically (d(Hi)- 1)-dimensional, * *and is acyclic if Hi2=F0. Also, dim(Xi-1Hi) d(Hi): this is clear if i = 0 (dim (X0) =* * 0), and holds for i 1 by (a) since d(Hj) d(Hi) for j < i by assumption. Thus, if Hi2=* * F0, we can set Xi= Xi-1. Assume now that Hi 2 F0. Write H = Hi and d=d(H) for short. If d = 1, then (Xi-1)H is 1-dimensional, and its connected components are all acyclic. By Lemm* *a 1.5, applied to the N(H)=H-set ss0((Xi-1)H ) (the set of connected components of (Xi* *-1)H ), (Xi-1)H has one connected component which is fixed by the action of N(H)=H, and* * the other components are permuted freely by N(H)=H. So Proposition 0.2(d) applies * *to Bob Oliver and Yoav Segev * * 13 show that there is a finite G-complex Xi, obtained by attaching orbits of cells* * G=Hx D1 to Xi-1, such that (Xi)H is acyclic. If d > 1, then by Proposition 0.2(a), there is a G-complex Y Xi-1, construc* *ted by attaching cells G=Hx Dk for 1 k d - 1, such that Y H is (d - 2)-connected and Hd-1(Y H) is Z-free. In particular, Y H is still homologically (d - 1)-dim* *ensional, and dim(Y H) d. For any subgroup 1 6= K=H N(H)=H of prime power order, (Y H)K=H = Y K = (Xi-1)K is acyclic by (d): K 2 F by definition of a separating* * family, and so K 2 Hi-1. Proposition 1.7 now applies to show that Hd-1(Y H) is stably f* *ree as a Z[N(H)=H]-module. So by Proposition 0.2(c), we can attach orbits of cells of * *type G=Hx Dk for k = d - 1; d to Y , to obtain a finite G-complex Xi Y such that (Xi* *)H_ is acyclic. |* *__| In fact, one can show for any family F of subgroups of G that there is a univ* *ersal (G; F)-complex. But such a complex must be infinite dimensional if F is not a s* *eparating family. We can now state necessary and sufficient conditions for the existence of uni* *versal or H-universal (G; F)-complexes of a given dimension. Proposition 1.9. Let G be any finite group, and let F be a separating family for G. Let F0 F be any subfamily with the property that N (F>H ) is contractible (and nonempty) for all H 2 Fr F0. Then there is a finite universal (G; F0)-comp* *lex. Furthermore, the following four conditions are equivalent for any m 2: (a) There exists an m-dimensional universal (G; F)-complex (H-universal if m * *= 2). (b) There exists a finite m-dimensional universal (G; F0)-complex (H-universa* *l if m = 2). (c) N (F>H ) is homologically (m - 1)-dimensional for each subgroup H 2 F0. (d) N ((F0)H ) is homologically (m - 1)-dimensional for each set H of subgro* *ups of G. Proof. Since the nerve N (F) is finite dimensional, the existence of a finite u* *niversal (G; F0)-complex follows from Proposition 1.8. (a ) d) If X is an m-dimensional H-universal (G; F)-complex, then for any se* *t of subgroups H, XH is homologically (m - 1)-dimensional by Lemma 1.6. Since H*(XH ) ~=H*(N (FH )) ~=H*(N ((F0)H )) by Proposition 1.3 and Lemma 1.4, N ((F0)H ) is also homologically (m-1)-dimen* *sional. (d ) c) Follows immediately from Lemma 1.4. (c ) b) Follows immediately from Proposition 1.8. * *__ (b ) a) Follows immediately from Lemma 1.4. |* *__| As an immediate corollary of Proposition 1.9, we get: Corollary 1.10. Let G be any finite group, and let F be a separating family for* * G. Then there is a (finite) 2-dimensional H-universal (G; F)-complex if and only i* *f N (F>H ) is homologically 1-dimensional for each subgroup H 2 F, if and only if N (FH )* * is__ homologically 1-dimensional for each set H of subgroups of G. * * |__| 14 Fixed point free actions on acyclic 2-complexes 2.Numbers of cells Again, G will always be a finite group throughout this section. We prove here* * some results which will be useful for keeping track of Euler characteristics of (uni* *ons of) fixed point sets in H-universal G-complexes. The notation used for doing this is defi* *ned as follows: Definition 2.1.For any family F of subgroups of G, define 1 iF (H) = i(G;F)(H) = _________. 1 - O(N (F>H )) : [N(H):H] for each H 2 F. Set I(G; F) = i(G;F)(1). We first note the following elementary relation between Euler characteristics* * of G- complexes and of their orbit spaces. Lemma 2.2. Let X0 X be any pair of finite G-complexes, and assume that all or* *bits in Xr X0 are of type G=H for some fixed subgroup H G. Then 0 O(X) - O(X0) = |G=H|. O(X=G) - O(X =G) : Proof.For each nP 0, let cn denote the number of n-cells in X not in X0. Then O(X) - O(X0) = n0 (-1)ncn. By assumption, each G-orbit of cells has order exa* *ctly |G=H|. So the number of n-cells in X=G not in X0=G is __1_|G=H|.cn for each n, * *and thus X cn 1 __ O(X=G) - O(X0=G) = ______ = ______ O(X) - O(X0) : |__| n0 |G=H| |G=H| The relation between these indices and Euler characteristics of universal com* *plexes is given in the following two lemmas. Lemma 2.3. Fix a separating family F, a finite H-universal (G; F)-complex X, * *and a subgroup H G. ForPeach n, let cn(H) denote the number of orbits of n-cells of * *type G=H. Then i(H) = n0 (-1)ncn(H). Proof.By Proposition 1.3, there is a G-map f : X ! N (F), which restricts to ho* *mology equivalences XH ! N (FH ) and X>H ! N (F>H ). Thus, by Definition 2.1, and * *by Lemma 2.2 applied to the action of N(H) on the complexes X>H XH , 1 1 H >H iF (H) = _________. 1 - O(N (F>H ))= _________. O(X ) - O(X ) [N(H):H] [N(H):H] = O(XH =N(H)) - O(X>H =N(H)): Each orbit of cells of type G=Hx Dn in X restricts to one of type (N(H)=H)x Dn * *in XH , and hence to exactly one n-cell in the orbit space XH =N(H). These are pre* *cisely the cells in XH =N(H) which are not in X>H =N(H), and hence X __ O(XH =N(H)) - O(X>H =N(H)) = (-1)ncn(H): |__| n0 Bob Oliver and Yoav Segev * * 15 Lemma 2.4. Let F be any separating family of subgroups of G, and let X be any finite H-universal (G; F)-complex. Let H F be any subset with the property th* *at K H 2 H and K 2 F implies K 2 H. Then H X O(N (H)) = O X = [N(H):H].iF (H): (1) H2H If, furthermore, H is a family (i.e., a union of G-conjugacy classes), then H X O X =G = iF (H): (2) H2H=conj Proof.We prove these formulas by induction on |H|; they clearly (vacuously) hol* *d when H = ;. Let H be a minimal subgroup of H, and set H0 = Hr {H}. Then N (H) = N (H0) [N(F>H) C(N (F>H )); in other words, the union of N (H0) and C(N (F>H ))* * (the cone over N (F>H )) with intersection N (F>H ). So by the Mayer-Vietoris sequen* *ce for the union, O(N (H)) = O(N (H0)) + 1 - O(N (F>H )) = O(N (H0)) + [N(H):H].iF (H); P and so O(N (H)) = H2H [N(H):H].iF (H) by induction. Since O(N (H)) = O(XH ) by Proposition 1.3, this proves (1). Now assume that H is a family. For each n 0 and each H 2 H, let cn(H) be the number of orbits of n-cells of type G=H. Let cn(H) be the sum, taken over conju* *gacy class representatives for all H 2 H, of the cn(H). Then cn(H) is precisely the * *number of n-cells in XH =G; and so H X1 n X X1 n X O X =G = (-1) cn(H) = (-1) cn(H) = iF (H) n=0 H2H=conjn=0 H2H=conj __ by Lemma 2.3. |__| Corollary 2.5. For any separating family F of subgroups of G, X iF (H) = 1: H2F=conj Proof.If X is any finite H-universal (G; F)-complex, then in particular X is ac* *yclic, and so X=G is acyclic (cf. [Br, Theorem III.7.12]). Thus O(X=G) = 1, and so the* *_result_ follows from Lemma 2.4 (applied with H = F). |* *__| The following relations will be useful later, when manipulating nerves of sub* *groups of G. Lemma 2.6. Fix a separating family F of subgroups of G. Let Fc F be the subf* *amily of those subgroups H 2 F such that N (F>H ) is not contractible. Fix a subgroup* * H 2 Fc such that H $ N(H) 2 F, and let K1; : :;:Kr be G-conjugacy class representative* *s for the subgroups K 2 Fc such that K % H and NK (H) = H. For each j, let aj be the number of Kj-conjugacy classes of subgroups in Kj which are G-conjugate to H and self-normalizing in Kj. Then Xr i(G;F)(H) = - aj.i(G;F)(Kj): (1) j=1 16 Fixed point free actions on acyclic 2-complexes Proof. For any subgroup H 2 Fr Fc, N (F>H ) is contractible, and so iF (H) = 0 by Definition 2.1. So we can assume that the K1; : :;:Kr contain G-conjugacy c* *lass representatives for all subgroups K 2 F such that K % H and NK (H) = H (not just those in Fc), without changing the right-hand side in (1). Let X be any finite H-universal (G; F)-complex. Set H = FH , and set H0 = {K* * 2 F | K H; NK (H) % H}. Then N (H) and N (H0) are both contractible by Lemma 0.3(b): the first since H has smallest element H; and the second since N(H) 2 H* *0, and N(H) \ K 2 H0 for all K 2 H0. By Lemma 2.4, X [N(K):K].iF (K) = O(N (H)) - O(N (H0)) = 1 - 1 = 0: (2) K2HrH0 Set R = {K 2 F | K % H; NK (H) = H}; the subgroups K1; : :;:Kr are thus G- conjugacy class representatives for the elements of R. For each j, set Sj = {g 2 G | gKjg-1 H; NgKjg-1(H) = H} = {g 2 G | gKjg-1 H; NKj(g-1Hg) = g-1Hg}: Then by (2), X |N(K)|.|H| X |N(K)| iF (H) = - __________iF (K)= - _________iF (K) K2R |N(H)|.|K| K2R0|N(H).K| 1 Xr X 1 = - @ ______________-1AiF (Kj); j=1 g2Sj|N(g Hg).Kj| and it remains only to show that the sum in parentheses is equal to aj: the num* *ber of Kj-conjugacy classes of subgroups g-1Hg for g 2 Sj. And this follows since f* *or any g; g0 2 Sj, g-1Hg and g-10Hg0 are Kj-conjugate if and only if there exists a 2 * *Kj such that a-1g-10Hg0a = g-1Hg, if and only if g-1g0a 2 N(g-1Hg) for some a 2 Kj, if_* *and_ only if g-1g0 2 N(g-1Hg).Kj. |_* *_| 3. Construction of 2-dimensional actions Again, in this section, G always denotes a finite group. To simplify the stat* *ements of results here and later, for any separating family F of subgroups of G, we wr* *ite (G; F) 2 U2 whenever there exists a 2-dimensional H-universal (G; F)-complex (a* *nd (G; F) =2U2 otherwise). We are now ready to construct the 2-dimensional acyclic actions of the groups* * G listed in Theorem A. But we first must look more closely at the question of which subg* *roups of G need not appear as isotropy subgroups in a universal (G; F)-complex. For any G and any separating family F of subgroups of G, we say that H 2 F is a critical subgroup in F if N (F>H ) is not contractible. As seen in Propositi* *on 1.9, subgroups which are not critical need not occur as isotropy subgroups in (H-) u* *niversal (G; F)-complexes. When notation is needed, we will denote by Fc the subfamily * *of Bob Oliver and Yoav Segev * * 17 critical subgroups in F. In the following lemma, we note some conditions which * *allow us to show that certain subgroups in F are not critical. Lemma 3.1. Let F be any family of subgroups of G which has the property that H H0 H00and H; H002 F imply H0 2 F. Fix a subgroup H 2 F. Then N (F>H ) ' * if any of the following conditions hold: (a) H is not an intersection of maximal subgroups in F. (b) There is a subgroup bH% H, bH2 F, such that K \ bH% H for all H $ K 2 Fc. Proof. (a) Let F0 F be the subfamily of all intersections of maximal subgroups in F, and let ff: F ! F0 be the function which sends a subgroup to the intersec* *tion of the members of Fmax which contain it. Then ff induces a deformation retract* *ion N (F>H ) ! N (F0>H) (Lemma 0.3(a)); and N (F0>H) is contractible since it conta* *ins the minimal element ff(H). (b) Set H = {K 2 F | K \ bH% H}. Then bH2 H, and K \ bH2 H for all K 2 H. So N (H) is contractible by Lemma 0.3(b). Now (Fc)>H = (Fc)H by assumption, and so N (F>H ) ' N ((Fc)>H ) = N ((Fc)H ) ' N (FH ) = N (H) ' *; * *__ where the homotopy equivalences follow from Lemma 1.4. |* *__| The following lemma provides a simple sufficient condition for the existence * *of a 2- dimensional H-universal (G; F)-complex. Lemma 3.2. Let F be any separating family of subgroups of G. Assume, for eve* *ry nonmaximal critical subgroup 1 6= H 2 F, that N(H) 2 F, and that K \ N(H) % H for all nonmaximal critical subgroups K % H in F. Then (G; F) 2 U2. More precisely, let M1; : :;:Mn be conjugacy class representatives for the ma* *ximal subgroups of F, and let H1; : :;:Hk be conjugacy class representatives for all * *nonmaximal critical subgroups of F. Then there is a 2-dimensional H-universal (G; F)-compl* *ex X which consists of one orbit of vertices of type G=Mi for each 1 i n, (-iF (H* *j))- orbits of 1-cells of type G=Hj for each 1 j k, and free orbits of 1- and 2-ce* *lls. If, furthermore, G is simple, then X can be constructed to contain exactly iF (1) f* *ree orbits of 2-cells (and no free orbits of 1-cells). Proof.Fix a nonmaximal critical subgroup H = Hj 2 F. If (Fc)>H Fmax, then N (F>H ) ' N ((Fc)>H ) is homologically 0-dimensional by Lemma 1.4. Otherwise,* * let H be the set of all K 2 F>H such that K \ N(H) % H, and set Hc = H \ Fc. Then N(H) 2 H, and K \ N(H) 2 H for all K 2 H, so N (H) is contractible (Lemma 0.3(b)). Since H F and Hc Fc are terminal subposets, Lemma 0.4 now applies to show that N (Hc) ' *. Thus, N ((Fc)>H ) consists of one contractible compon* *ent N (Hc), together with some isolated vertices for those maximal subgroups M 2 F>H such that M \ N(H) = H. In particular, N (F>H ) is homologically 0-dimensional. Hence, by Proposition 1.8, there is a finite H-universal (G; Fc)-complex X su* *ch that dim(XM ) = 0 for each maximal subgroup M 2 F, such that dim(XH ) = 1 for each nonmaximal subgroup 1 6= H 2 Fc, and such that each vertex of X is fixed by a m* *aximal subgroup in F. But by Proposition 1.3 and Lemma 1.4, H*(Xs) ~= H*(N (F>1)), so 18 Fixed point free actions on acyclic 2-complexes N (F>1) is homologically 1-dimensional since Xs is; and by Proposition 1.8 agai* *n, X can be taken to be 2-dimensional. By the above description of X, we see that all orbits of vertices in X are of* * type G=M for maximal M; that all orbits of edges are of type G=Hifor 1 i k or (possibl* *y) free (of type G=1); and that all orbits of 2-cells are free. Hence the numbers of or* *bits of cells of type G=Mi or G=Hj follows from the formula in Lemma 2.3. (Note that iF (M) =* * 1 whenever M is maximal.) Also, by Proposition 1.7, H1(Xs) is stably free as a Z* *[G]- module, and hence is free by Proposition C.4 if G is simple. So by Proposition * *0.2(c), X can be constructed by attaching only free orbits of 2-cells to Xs; and the nu* *mber_of orbits of cells is again given by Lemma 2.3. * * |__| Lemma 3.2 will be applied to construct 2-dimensional actions of the simple gr* *oups L2(q) (= P SL2(q)) for certain q, and of the Suzuki groups. We first list some* * of the properties of subgroups of the L2(q) which will be needed here, and also later * *in Section 6. Proposition 3.3. Fix q = pk 4, where_p is prime. Then the maximal solvable subgroups H L2(q) = P SL2(q) and H P GL2(q) are as described in the following table. (Note that L2(q) = P GL2(q) when q is a power of 2.) _____________________________________________________________ || H L2(q) (q odd)|| H P GL2(q) || || ||____________________||__________________||________________||_ || H | Nr. classes|| H | Nr. classes|| conditions || ||_________|__________||_______|__________||________________ || || Fo C(q-1)=2| 1 || Fo Cq-1| 1 || __ || ||_________|__________||_______|__________||_______________ || || Dq-1 | 1 || D2(q-1)| 1 || __ || ||_________|__________||_______|__________||_______________ || || Dq+1 | 1 || D2(q+1)| 1 || __ || ||_________|__________||_______|__________||_______________ || || A4 | 1 || 4 | 1 || q 3 (mod 8) || ||_________|__________||_______|__________||_______________|| || 4 | 2 || 4 | 1 || q 1 (mod 8) || ||_________|__________||_______|__________||_______________||_ __ Here, in all cases (when q is odd), H = NPGL2(q)(H). Furthermore, each nonsol* *vable subgroup of L2(q) is conjugate in P GL2(q) to one of the groups L2(q0) for q0 =* * pk0and k0|k; or to P GL2(q0) for q0 = pk0 and 2k0|k; or (if q is odd and q 1 (mod 5))* * is isomorphic to A5. Proof.See [Sz2, x3.6]. The subgroups of L2(q) are described in [Sz2, Theorems 3* *.6.25- 26], and in [H1 , 8.27]. The uniqueness up to conjugacy of the dihedral groups * *follows from [Sz2, 3.6.23]; and the uniqueness of the Fqo Cq-1 or Fqo C(q-1)=2follows s* *ince they are normalizers of Sylow p-subgroups. The maximal subgroups A4 or 4 are normali* *zers of elementary abelian subgroups (C2)2 L2(q), of which there is one or two conj* *ugacy classes depending on q (mod 8) (see also [H1 , 8.16]). The fact that any subgr* *oup isomorphic to L2(q0) or P GL2(q0) is conjugate (in P GL2(q)) to the standard on* *e follows from [Sz2, 3.6.20 and Ex. 3.6.1+3]. Note in particular that B (~=Fqo Cq-1or ~=Fqo C(q-1)=2) is represented by the* * group of upper triangular matrices, and that D2(q-1)is the subgroup of monomial matrices* *. The other dihedral group D2(q+1)or Dq+1is the subgroup of GL(Fq2) (here Fq2is viewe* *d as a 2-dimensional vector space over Fq) of all transformations of determinant one g* *enerated Bob Oliver and Yoav Segev * * 19 by multiplying by an element of Fq2or by applying the Frobenius automorphism (x* * 7! xq). Finally, the results about maximal subgroups of P GL2(q) follow from the info* *rmation_ about subgroups of L2(q2) P GL2(q). |_* *_| We first construct actions of the groups L2(2k). Example 3.4. Set G = L2(q), where q = 2k and k 2. Then there is a a 2-dimensi* *onal acyclic fixed point free G-complex X, all of whose isotropy subgroups are solva* *ble. More precisely, X can be constructed to have three orbits of vertices with isotropy * *subgroups isomorphic to Fqo Cq-1, D2(q-1), and D2(q+1); three orbits of edges with isotro* *py sub- groups isomorphic to Cq-1, C2, and C2; and one free orbit of 2-cells. Proof. Let SLV be the separating family of solvable subgroups of G, and let SL* *V c SLV be the subfamily of all critical subgroups in SLV . By Proposition 3.3, th* *e maximal solvable subgroups of G are the groups B = Fqo Cq-1, D2(q-1), and D2(q+1), wher* *e each occurs with exactly one conjugacy class. The Borel subgroups of G are those conjugate to B; or equivalently those subg* *roups of G which fix a line (a 1-dimensional subspace of (Fq)2). Every subgroup of G* * of even order is contained in at most one Borel subgroup, since the subgroup of el* *ements fixing any two distinct lines is cyclic of order q- 1. Also, any subgroup cont* *ained in both a Borel subgroup and a dihedral subgroup must have order 2. Thus, C2 is t* *he only subgroup of even order contained in more than one maximal subgroup in SLV . Any nontrivial odd order subgroup is contained in a unique maximal dihedral sub* *group (its normalizer); and a subgroup Cr for 1 6= r|(q- 1) is contained in exactly t* *wo Borel subgroups corresponding to the two lines (eigenspaces) it leaves invariant. Thu* *s, since each critical subgroup must be an intersection of maximal subgroups in SLV (Le* *mma 3.1), the only possible critical subgroups are the maximal subgroups, together * *with Cq-1, C2, and 1 (one conjugacy class each). Computations using Lemma 2.6 (and Corollary 2.5 to determine iSLV(1)) now yie* *ld the following table. _____________________________________ || H 2 SLV c| K \ N(H) = H | i(H) || ||____________|______________|______||_ || B = Fqo Cq-1| _ | 1 || ||____________|______________|_____||_ || D2(q-1) | _ | 1 || ||____________|______________|_____||_ || D2(q+1) | _ | 1 || ||____________|______________|_____||__ || Cq-1 | B | -1 || ||____________|______________|______|| || C2 | D2(q 1) | -2 || ||____________|______________|______||_ || 1 | _ | 1 || ||____________|______________|_____||__ Table 1 20 Fixed point free actions on acyclic 2-complexes From this, it is clear that the hypotheses of Lemma 3.2 are satisfied, and henc* *e that (L2(q); SLV ) 2 U2. More precisely, the lemma and table show that there is an * *H- universal (G; SLV c)-complex, with three orbits G=B, G=D2(q-1), and G=D2(q+1)of* * ver- tices; with three orbits G=C2, G=C2, and G=Cq-1 of 1-cells; and with one free o* *rbit_of 2-cells. |_* *_| Before continuing with the construction of the actions of other groups, we wa* *nt to discuss the classical example of an A5-action, and its relationship with the co* *nstruction (when G = L2(4) ~= A5) in Example 3.4. We first establish our notation. We wr* *ite SO(3) = SO(3; R), and write S3 = SL1(H) ~=SU(2; C) for the group of unit quater* *nions (elements of norm one in the quaternion algebra H over R). There is a homomorph* *ism S3 ! SO(3), surjective with kernel {1}, which is defined by sending a 2 S3 H to the matrix of the conjugation map (x 7! axa-1) on the subspace H. Th* *us, we regard S3 as a two fold cover of SO(3). We now identify A5 ~=L2(5) as the icosahedral subgroup of SO(3), and let A*5~= SL2(5) (the binary icosahedral group) denote its inverse image in S3. Consider* * the action of A5 via left multiplication on the space 3 = SO(3)=A5 ~=S3=A*5of left * *cosets. This space is the Poincare sphere, a 3-manifold which has the homology of the 3* *-sphere, and whose fundamental group is isomorphic to the perfect group A*5. Then A5 act* *s with fixed point set (SO(3)=A5)A5 = N(A5)=A5 = pt. Upon removing an open invariant b* *all around the fixed point, we obtain a compact acyclic 3-manifold M (with boundary* *) upon which A5 acts without fixed points. This was the starting point for the constru* *ction by Floyd and Richardson [FR ] of an action of A5 on a disk without fixed points (s* *ee also [Br, xI.8] for more details). Since @M 6= ;, M can now be collapsed to a 2-dime* *nsional subcomplex X ' M, upon which A5 still acts without fixed point. This last step can be made more explicit. Let P denote the regular polytope w* *ith 120 dodecahedral faces, and let be its symmetry group. Clearly, SO(4) ~=S3x C2S3, and contains A5 (the group of symmetries leaving one face invariant) with index 120. This implies that ~=A*5xC2A*5; and hence that contains a binary icosahed* *ral subgroup A*5which permutes freely the faces of P . So 3 ~=S3=A*5can be identif* *ied with the space D=~ , obtained by identifying opposite faces of the solid dodeca* *hedron D in an appropriate way. This is in fact Poincare's original construction of th* *e Poincare sphere. For more details on the identification, and another way of showing that* * these two constructions are equivalent, we refer to [KS , pp. 124-128]. Under this identification of 3 with D=~ , the A5 action on 3 is induced by th* *e usual action on the dodecahedron. The fixed point is thus the center of D; and the op* *eration of removing the fixed point and collapsing the remaining space to a 2-dimension* *al sub- complex corresponds to removing the center of D and then collapsing to its boun* *dary. The result is an explicit 2-dimensional complex X = @D=~ with fixed point free * *action of A5, which has 6 pentagonal 2-cells, 10 edges, and 5 vertices. Here's another, quicker way to construct this last complex. Let X0 be the 1-s* *keleton of the 4-simplex, with the obvious action of A5 permuting the five vertices. An* *y 5-cycle in A5 (in the vertices of X0) tells us how to attach a pentagon to X0; and two * *such pentagons will be in the same orbit of A5 if and only if the corresponding 5-cy* *cles are conjugate. So by attaching to X0 six pentagons corresponding to one conjugacy c* *lass of 5-cycles in A5, we obtain a 2-complex X with A5-action. One can check directly * *that Bob Oliver and Yoav Segev * * 21 X is acyclic (and with a bit more work show that ss1(X) ~=A*5); but one also se* *es easily that it is identical with the previous construction based on the dodecahedron. If we now subdivide each pentagon in (either of) these spaces, as a union of * *ten 2-simplices (by adding extra vertices at the midpoints of edges and centers of * *faces), we have constructed an A5-complex of the type constructed in Example 3.4 _ exce* *pt that the 2-cells have been attached explicitly. This is also identical to the A* *5-simplicial complex constructed in [S1, x3] and in [AS , x9]. We also note here that for k * * 3, the L2(2k)-complexes constructed in Example 3.4 have the same 1-skeleton as the com* *plexes constructed in [AS , x9] (which were not acyclic); they differ only in the way * *the 2-cells are attached. We now consider G = L2(q), when q 3 (mod 8) is an odd prime power. Example 3.5. Assume that G = L2(q), where q = pk 5 and q 3 (mod 8). Then there is a a 2-dimensional acyclic fixed point free G-complex X, all of whose i* *sotropy subgroups are solvable. More precisely, X can be constructed to have four orbi* *ts of vertices with isotropy subgroups isomorphic to Fqo C(q-1)=2, Dq-1, Dq+1, and A4* *; four orbits of edges with isotropy subgroups isomorphic to C(q-1)=2, C22, C3, and C2* *; and one free orbit of 2-cells. Proof. Since L2(5) ~=L2(4) has already been dealt with in Example 3.4, we assum* *e for simplicity that q > 5. Let SLV be the separating family of solvable subgroups* * of G, and let SLV c SLV be the subfamily of all critical subgroups in SLV . By Propo* *sition 3.3, the maximal solvable subgroups of G are the groups Dq-1; Dq+1; A4; and B = Fqo C(q-1)=2; where each occurs with exactly one conjugacy class. Any subgroup H 2 SLV of order a multiple of p is contained in a unique subgr* *oup conjugate to B (it fixes a unique line in (Fq)2); and is contained in one of th* *e other maximal subgroups only if p = 3 and H ~=C3. If 1 6= H 2 SLV has order prime to* * p, is not maximal, and is not isomorphic to C2, then either it is cyclic of order * *dividing (q - 1)=2 and contained in one dihedral group and two Borel subgroups (correspo* *nding to the two lines in (Fq)2 fixed by H), or it is cyclic of order dividing (q + 1* *)=2 and is contained in a unique Dq+1 (its normalizer), or H is dihedral and contained in * *a unique maximal dihedral subgroup Dq1 (the normalizer of its index 2 subgroup). Since * *each critical subgroup must be an intersection of maximal subgroups in SLV (Lemma 3* *.1), we have now shown that the only possible critical subgroups are the maximal sub* *groups, together with one conjugacy class each of subgroups C(q-1)=2, C3, C22, C2, and * *1. In the following table, D+ denotes the maximal dihedral subgroup of order q * *1 0 (mod 4), and D- the other (conjugacy class of) maximal dihedral subgroup (note * *that D+ = N(C2)). Recall that we are assuming that q > 5 (otherwise Dq-1= C22). 22 Fixed point free actions on acyclic 2-complexes _______________________________________ || H 2 SLV c | K \ N(H) = H | i(H) || ||_______________|______________|______|| || B = Fqo C(q-1)=2| _ | 1 || ||_______________|______________|_____|| || Dq-1 | _ | 1 || ||_______________|______________|_____|| || Dq+1 | _ | 1 || ||_______________|______________|_____|| || A4 | _ | 1 || ||_______________|______________|_____||_ || C(q-1)=2 | B | -1 || ||_______________|______________|_____ || || C2 | D+ | -1 || ||________2______|______________|_____ || || C3 | A4 | -1 || ||_______________|______________|_____ || || C2 | D- | -1 || ||_______________|______________|______|| || 1 | _ | 1 || ||_______________|______________|_____||_ Table 2 As before, the computations of iSLV(H) for nonmaximal 1 6= H G all follow from Lemma 2.6, and the computation of iSLV(1) then follows from Corollary 2.5. Lemma 3.2 now applies to show that (L2(q); SLV ) 2 U2. More precisely, toget* *her with Table 2, it shows that a 2-dimensional H-universal (L2(q); SLV )-complex X* * can be constructed with four orbits of vertices of types G=B, G=Dq-1, G=Dq+1, and G* *=A4; four orbits of 1-cells of types G=C22, G=C(q-1)=2, G=C3, and G=C2; and one free* * orbit of 2-cells. Note that G=C(q-1)=2xD1 always connects the orbits G=B and G=Dq-1, * *and G=C2x D1 always connects the orbits G=Dq-1and G=Dq+1. The orbit of cells G=C22x* *D1 connects G=A4 to G=Dq-1 or G=Dq+1, depending on q modulo 8. And the orbit of ce* *lls G=C3x D1 connects G=A4 to one of G=B (if q = 3k), or to G=Dq1 (whichever has or* *der_ a multiple of 3). |* *__| The third family of groups with 2-dimensional actions consists of the Suzuki * *groups Sz(q), for all q = 22k+1 8. In order to specify more precisely subgroups of S* *z(q), we regard it as a subgroup of GL4(Fq) as described in [HB3 , xXI.3]. The follo* *wing properties of Sz(q) and its subgroups will be needed here, as well as in Sectio* *n 6. Proposition 3.6. Fix q = 22k+1, and let 2 Aut (Fq) be the automorphism x = k+1 p__2q 2 * x2 = x (thus (x ) = x ). For a; b 2 Fq and 2 (Fq) , define elements 0 1 1 0 0 0 B a 1 0 0C S(a; b) = B@ b a 1 0CA; a2+ + ab + b a1+ + b a 1 and 0 k 1 0 1 1+2 0 0 0 0 0 0 1 B 0 2k 0 0 C B 0 0 1 0C M() = B@ -2k C ; o = B C : 0 0 0 A @ 0 1 0 0A k 0 0 0 -1-2 1 0 0 0 Set S(q; ) = , T = ~= Cq-1, and B = M(q; ) = S(q; )o T and N = ~=D2(q-1): Bob Oliver and Yoav Segev * * 23 Then Sz(q) ~=, and under this identification the following hold: (a) S(q; ) is a Sylow 2-subgroup of Sz(q). (b) There are four conjugacy classes of maximal subgroups in Sz(q) which are * *solvable: (B), (N), (M+), and (M-), where M+ ~=Cq+p__2q+1oC4 and M- ~=Cq-p__2q+1oC4: These are the only maximal solvable subgroups in Sz(q). (c) Each nonsolvable subgroup of Sz(q) is conjugate to Sz(q0), for some q0 = * *22m+1 where (2m + 1)|(2k + 1). (d) Sz(q) is contained in the 4-dimensional symplectic group over Fq: Sz(q) Sp4(q) def={g 2 GL4(q) | gogt = o}; where gt is the transpose of g and o is as above. (e) All of the subgroups B; N; T; S(q; ); Sz(q) are invariant under the autom* *orphisms of GL4(q) induced by automorphisms of the field Fq. p __ p __ (f) | Sz(q)| = q2(q- 1)(q2+ 1) = q2.(q- 1).(q+ 2q+1).(q- 2q+1), where the* * four fac- tors in the second expression are pairwise relatively prime. Proof.See [HB3 , xXI.3]. Note in particular the relations S(a; b).S(c; d) = S(a + c; b + d + a c) and M()-1S(a; b)M() = S(a; 1+ b): The list of maximal subgroups of Sz(q) (points (b) and (c)) is shown in [Sz1, T* *heorem 9]. Note that if q0 = 22m+1, where (2m + 1)|(2k + 1), then Sz(q) \ GL4(q0) = Sz(q* *0) (and similarly for the other subgroups). The inclusion Sz(q0) Sz(q) follows since 2* *k 2m k 2m (mod 22m+1- 1), and hence x2 = x for all x 2 Fq0. The inclusion Sz(q) \ GL4(q* *0)_ Sz(q0) then follows from (c). * *|__| We are now ready to construct actions of Sz(q) on acyclic 2-complexes. Example 3.7. Set q = 22k+1, for any k 1. Then there is a 2-dimensional acycli* *c fixed point free Sz(q)-complex X, all of whose isotropy subgroups are solvable. More * *precisely, X can be constructed to have four orbits of vertices with isotropy subgroups is* *omorphic to M(q; ), D2(q-1), Cq+p__2q+1oC4, and Cq-p__2q+1oC4; four orbits of edges with* * isotropy subgroups isomorphic to Cq-1, C4, C4, and C2; and one free orbit of 2-cells. Proof. Set G = Sz(q). By Proposition 3.6, G contains the following maximal solv* *able subgroups: M(q; ); D2(q-1); Cq+p__2q+1oC4; and Cq-p__2q+1oC4; with one conjugacy class for each isomorphism type. If 1 6= H 2 SLV and (|H|; * *q2+1) 6= 1, then H is contained in a unique maximal subgroup Cq2p__2q+1oC4: the normaliz* *er of its unique maximal odd order subgroup. Likewise, if H is dihedral of order d* *ividing 2(q-1)f(andi|H| 6= 2), then H is contained in a unique maximal subgroup D2(q-1)* *; while if |H|fi(q - 1) then H is contained in the same maximal subgroups as its centra* *lizer of order q - 1. Any subgroup of even order which is not dihedral is contained in a* *t most one maximal subgroup, conjugate to M(q; ). (The centralizer of any involution i* *n G is 24 Fixed point free actions on acyclic 2-complexes a 2-group by [Sz1, Proposition 1], and each involution in the Sylow subgroup S(* *q; ) is central. So an involution cannot be in two Sylow subgroups.) Thus, any subgroup* * which is an intersection of two or more maximal subgroups is isomorphic to one of the* * groups Cq-1, C4, C2, or 1; and these are the only possible critical subgroups by Lemma* * 3.1(a). There is just one conjugacy class each of subgroups Cq-1 or C2 (note, for examp* *le, that all subgroups of order 2 in S(q; ) are conjugate in M(q; )). By [Sz1, Propositi* *on 18], G contains two conjugacy classes of elements of order 4, and it is easy to chec* *k by direct calculations that an element of order 4 in G is not conjugate to its inverse. H* *ence G contains just one conjugacy class of C4's. Now let SLV cbe the subfamily of critical subgroups in SLV . Consider the fol* *lowing table of values for iSLV(H) for H 2 SLV c: _____________________________________ || H 2 SLV c| K \ N(H) = H | i(H) || ||____________|______________|______||_ || B = M(q; )| _ | 1 || ||____________|______________|_____||_ || D2(q-1) | _ | 1 || ||____________|______________|_____||_ || C p__ o C4| _ | 1 || ||__q+_2q+1___|______________|_____||_ || C p__ o C4| _ | 1 || ||__q-_2q+1___|______________|_____||__ || Cq-1 | M(q; ) | -1 || ||____________|______________|______|| || C4 | C p__ o C4| -2 || ||____________|___q_2q+1_____|______|| || C2 | D2(q-1) | -1 || ||____________|______________|______||_ || 1 | _ | 1 || ||____________|______________|_____||__ Table 3 When H ~=Cq-1, C4, or C2, then iSLV(H) is computed using Lemma 2.6. (Note that C2 can never be self-normalizing in any group of order a multiple of 4.) The va* *lue of iSLV(1) then follows from Corollary 2.5. Lemma 3.2 now applies to show that (Sz(q); SLV ) 2 U2. More precisely, there * *is a 2-dimensional H-universal (Sz(q); SLV )-complex which has four orbits of vertic* *es and four orbits of edges (with isotropy subgroups as given in Table 3), and one fre* *e_orbit of 2-cells. |_* *_| 4. Reduction to simple groups Throughout this section, G will be a finite group. Recall that a G-complex X * *is called essential if there is no normal subgroup 1 6= N C G, with the property that the* * inclusion XN X is a G-Z-equivalence; i.e., such that XNH ! XH is a homology equivalen* *ce for all H G. We would like to be able to show directly that all groups which h* *ave essential fixed point free actions on acyclic 2-complexes are simple. Instead,* * in this section, we prove a slightly weaker result (Proposition 4.4), where we show tha* *t any group with such an action is an extension of a simple group by outer automorphi* *sms. Bob Oliver and Yoav Segev * * 25 The proof of this uses the result in [S1] that the fixed point set of any gro* *up acting on a 2-dimensional acyclic complex must be acyclic or empty. Since the proof in* * [S1] requires the odd order theorem, we give here a different one, which is more ele* *mentary. Theorem 4.1 [S1, Theorem 3.4].Let X be any 2-dimensional acyclic G-complex (not necessarily finite). Then XG is acyclic or empty, and is acyclic if G is solvab* *le. Proof.The first half of the following proof is essentially the same as that in * *[S1], but is included here for the sake of completeness. If G is a p-group for some prime p, then XG is Z=p-acyclic by Smith theory (c* *f. [Br, Theorem III.7.12]), and homologically 1-dimensional by Lemma 1.6. It follows th* *at XG is Z-acyclic in this case. Now assume that G is a minimal group for which there is a counterexample. The* *n G must be simple and nonabelian _ since if N C G were a proper normal subgroup, t* *hen XN would be acyclic, and hence XG = (XN )G=N would be acyclic or empty (acyclic if G isTsolvable) by the minimality of G. Also, XH is acyclic for all H $ G, * *and XG = H$G XH is homologically 0-dimensional by Lemma 1.6 again. In other word* *s, each connected component of XG is acyclic, and it remains to show that there is* * at most one component. Assume otherwise: let k 2 be the number of connected components of XG . Let F be the (separating) family of proper subgroups H $ G. Very roughly, we will sho* *w that X "looks like" the join of an H-universal (G; F)-complex Y with a set of k poin* *ts. But for X to be 2-dimensional, Y would have to be 1-dimensional, i.e., a tree; and * *this is impossible. To make this precise, let F+ denote the poset which consists of F, together w* *ith k elements (G; i) for i = 1; : :;:k. Extend the ordering on F by setting (G; i) * *H for all H 2 F, and with no inclusion relations between the (G; i). Write XG = F1 q . .q* *.Fk, where the Fi are the connected components. We now apply Lemma 0.1, with the covering of X given by XH = XH for H 2 F, and X(G;i)= Fi. Thus, Xffis acyclic * *for each ff 2 F+. So by Lemma 0.1, for each H 2 F, H*(X>H ) ~=H*(N ((F+)>H )), and thus N ((F+)>H ) is homologically 1-dimensional (Lemma 1.6). But the poset (F+)* *>H consists of F>H together with the elements (G; i), and so its nerve is the uni* *on of k cones over N (F>H ). This complex contains the suspension of N (F>H ) as a retr* *act (i.e., the case k = 2); and hence N (F>H ) is homologically 0-dimensional. Since this* * holds for all H 2 F, Proposition 1.8 now applies to show that there is a finite 1-dim* *ensional universal (G; F)-complex Y . But then Y is a tree upon which G acts without f* *ixed __ points, and this is impossible (cf. [Se, xI.6]). * * |__| The following easy consequence of Theorem 4.1 turns out to be very useful. It* *s proof involves collapsing out certain subcomplexes of a CW complex to create new fixe* *d points, and get a contradiction to Theorem 4.1. In general, if X is a G-complex and A X is a G-invariant subcomplex, then X=A is defined to be the quotient space X=~ ,* * where x~ y if x = y or x; y 2 A. This quotient space has an obvious structure as a G-* *complex: where (X=A)(n)= X(n)=~ , and where X=A has one vertex for the identification po* *int A=A and otherwise one cell for each cell in X not in A (see [LW , Theorem II.5.* *11], taking Y = pt). The homology groups of X, A, and X=A are linked by exact sequen* *ces 26 Fixed point free actions on acyclic 2-complexes (coming from the fact that Cn(X=A)=Cn(pt) ~=Cn(X)=Cn(A)). In particular, if A * *is acyclic, then H*(X=A) ~=H*(X). Corollary 4.2. Let X be any 2-dimensional acyclic G-complex. Assume that A; B X are G-invariant acyclic subcomplexes such that A [ B XG . Then A \ B 6= ;. Proof.Assume otherwise: that A \ B = ;. Let Y be the G-complex obtained by identifying the subcomplexes A and B each to a point. Then Y is still acyclic* *, since A and B are, and Y G consists of the two identification points. And this contr* *adicts_ Theorem 4.1, which says that Y G must be acyclic or empty. * * |__| As immediate consequences of Corollary 4.2 we get: Lemma 4.3. Let X be a 2-dimensional acyclic G-complex. Then the following hol* *d. (a) [AS , 4.5] If H; N G are such that H NG(K) and XH and XK are nonempty, then XHK 6= ;. (b) If H G is such that XH = ;, then XCG(H) 6= ;. Proof. If XG 6= ;, then (a) and (b) are obvious. So assume XG = ;. (a) Since H normalizes K, both XH and XK are H-invariant acyclic subcomplex* *es of X. So by Corollary 4.2, if XH and XK are nonempty, then XH \ XK = XHK 6= ;. (b) It suffices to prove this when H is minimal among subgroups without fixed* * points. Fix a pair M; M0 H of distinct maximal subgroups (H is nonsolvable). Then XM 0 M M0 H M M0 and XM are nonempty, but X \ X = X = X = ;. Thus X and X are disjoint CG(H)-invariant acyclic subcomplexes of X, and so CG(H) must have * *fixed_ points by Corollary 4.2. |* *__| As a first consequence of Lemma 4.3, we can now prove: Theorem B. Let G be any finite group, and let X be any 2-dimensional acyclic * *G- complex.0Let N be the subgroup generated by all normal subgroups N0 C G such th* *at XN 6= ;. Then XN is acyclic; X is essential if and only if N = 1; and if N 6=* * 1 then the action of G=N on XN is essential. Proof.If XN1 6= ; and XN2 6= ;, where N1; N2 C G, then X6= ; by Lemma 4.3(a). Thus XN is nonempty, and is acyclic by Theorem 4.1. The action of G=N * *on XN is always essential, since any nontrivial normal subgroup of G=N has empty * *fixed point set. Now assume that N 6= 1. For all H G, XH and XNH are acyclic or empty by Theorem 4.1; and XNH is nonempty if XH is by Lemma 4.3(a). So the inclusion XN* *H ! __ XH is always an equivalence of integral homology, and hence X is not essential.* * |__| We are now ready to prove: Proposition 4.4. If G is a nontrivial finite group for which there exists an es* *sential 2-dimensional acyclic G-complex X, then G is almost simple. More precisely, the* *re is a normal subgroup L C G, such that L is simple, such that XL = ;, and such that CG(L) = 1 (i.e., G Aut(L)). Bob Oliver and Yoav Segev * * 27 Proof.By Theorem B, XN = ; for all proper normal subgroups 1 6= N C G. In particular, XG = ;. Fix a minimal normal subgroup 1 6= L C G. Then L is nonsolvable, since XL 6= * *;. Hence L is a direct product of isomorphic nonabelian simple groups (cf. [Go , T* *heorem 2.1.5]). Assume first that L is not simple. By Lemma 4.3(b), XH 6= ; for some simple factor H C L; and L = since it is a minimal normal subgroup. Si* *nce -1 H L XgHg = g(X ) 6= ; for all g, X 6= ; by Lemma 4.3(a) (applied to the action * *of L on X). And this is a contradiction. Thus, L is simple. Set H = CG(L). Then H C G, and XH 6= ; by Lemma 4.3(b); * * __ and so H = 1 (again since the G-action on X is essential). * * |__| Using Proposition 4.4, when determining which finite groups have essential fi* *xed point free actions on 2-dimensional acyclic complexes, it suffices first to determine* * which simple groups have such actions, and then consider automorphism groups only of those s* *imple groups which do have them. 5. Some conditions for nonexistence of 2-dimensional actions Again, throughout this section, G is a finite group. We recall two definition* *s intro- duced in Section 3. If F is a separating family for G, then Fc denotes the sub* *family of critical subgroups for F: the set of all H 2 F such that N (F>H ) 6' *. An* *d U2 denotes the class of pairs (G; F) (where F is a separating family for G) for wh* *ich there exists a 2-dimensional H-universal (G; F)-complex. We have already constructed * *some examples of pairs (G; F) which do lie in U2, and next want to show that they ar* *e the only ones. In this section, we develop some general techniques for doing this. For any G-complex X, and any n > 1, it will be convenient to write X[n]to den* *ote the union of fixed pointfsetsiof subgroups of order a multiple of n; or equival* *ently the set of all x 2 X for which nfi|Gx|. Also, for any family F of subgroups of G, we wr* *ite F[n]to denote the subfamily of those subgroups in F of order a multiple of n. We will * *see that if (G; F) 2 U2, then not only is N (F[n]) homologically 1-dimensional for all n* *, but its orbit space N (F[n])=G is homologically 0-dimensional (i.e., its connected comp* *onents are acyclic). In Section 5a, conditions are established which allow us to directly detect e* *lements in H2(N (F[n])), for appropriate n, via Euler characteristic arguments. The pro* *perties of N (F[n])=G are shown in Section 5b, and then another set of criteria are fou* *nd which detect elements in H1(N (F[n])=G). Afterwards, conditions on G and F are set u* *p in Section 5c which imply that for any 2-dimensional H-universal (G; F)-complex X,* * the singular set Xs is itself acyclic (and hence H-universal); and then Section 5d * *deals with the problem of showing that this is impossible. 28 Fixed point free actions on acyclic 2-complexes 5a. Detecting_2-cycles_in_nerves_of_posets_of_subgroups_ Our main tool here for directly detecting elements in the second homology of * *nerves of posets of subgroups will be certain "coset complexes". We adopt the following n* *otation: Definition 5.1.Fix any group G, and any triple K1; K2; K3 of subgroups of G. De* *fine K1; K2; K3 = K1; K2; K3 G to be the G-simplicial complex with vertex set (G=K1) q (G=K2) q (G=K3) (where G acts by left translation), and with a 1- or 2-simplex for every pair or triple * *of cosets with nonempty intersection. Thus, each edge in K1; K2; K3 has the form [aKi; aKj] for some a 2 G and * *some 1 i < j 3, and each 2-simplex has the form [aK1; aK2; aK3] for some a 2 G. In many cases, one can show that H2( K1; K2; K3 ) 6= 0 via an easy counting argu* *ment: Lemma 5.2. Fix any group G, and any sequence K1; K2; K3 of subgroups of G. S* *et Kij= Ki\Kj, K = K1\K2\K3, and G0= . Assume that ___1___ 1 1 + _______+ _______ 1; (1) [K12:K] [K13:K] [K23:K] or (more generally) that X 1 X3 1 1 _______< 1 + ______ - ______: (2) 0:K] i; and so t* *here are exactly [G:G0] connected components. By definition, X has three orbits of vertices of type G=Ki, three orbits of e* *dges of type G=Kij, and one orbit of 2-simplices of type G=K. Hence X X3 O(X) = [G:K] - [G:Kij] + [G:Ki] i [G:G ] = rk(H0(X)); i. iK2, and * *let Q0be the set of those H 2 Q such that the coefficient in z of {K2; H; K0} is nonzero* *. By construction, every element of Q0 is G-conjugate (in fact, N02-conjugate) to K1* *; and by condition (c), every element of Q in the same N (Q)-connected component as K* *1 is contained in K1. Lemma 0.5 now implies that 0 6= [z] 2 H2(N (H)); and so (G; F)* *_=2U2_ by Proposition 1.9. |* *__| Two n-tuples of subgroups (H1; : :;:Hn) and (H01; : :;:H0n) in G will be call* *ed G- conjugate if there is some g 2 G such that H0i= gHig-1 for all i. The normaliz* *er NG(H1; : :;:Hn) of such an n-tuple is just the intersection of the normalizers * *NG(Hi). The next proposition is a somewhat more complicated application of Lemma 5.2. Proposition 5.4. Fix a separating family F of G. Let K1; K2; K3 2 F be three subgroups such that neither K2 nor K3 is conjugate to K1. Set Kij= Ki\ Kj and K = K1 \ K2 \ K3. Let F0 F denote the subfamily consisting of Fc, together with all subgroups conjugate to any of the Ki, Kij, or K. Assume the following condi* *tions hold: 1 1 1 (a1) _______ + _______+ _______ 1; or more generally [K12:K] [K13:K] [K23:K] 1 1 1 1 1 1 1 (a2) _______+ _______+ _______< 1 + _______+ _______+ _______- ______, where [K12:K] [K13:K] [K23:K] [K1:K] [K2:K] [K3:K] [G0:K] G0= . (b) K1 is maximal in F. (c) There is no H 2 F0 such that K $ H $ K12or K12$ H $ K1. (d) NG(K1; K12; K) = K. (e) The triples (K1; K12; K) and (K1; K13; K) are not G-conjugate. Then H2(F(K) ) 6= 0; and so (G; F) =2U2. Proof.Consider the complex X = K1; K2; K3 of Definition 5.1, and let X* den* *ote its barycentric subdivision. To distinguish between simplices of X* and of N (* *F), we put parentheses (-) around the former and curly brackets {-} around the latter.* * The vertices in X* will be denoted (gKi) (the vertices in X), (gKij) (the midpoint * *of the edge (gKi; gKj)), and (gK) (the barycenter of the 2-simplex (gK1; gK2; gK3)). 30 Fixed point free actions on acyclic 2-complexes We have H2(X) 6= 0 by (a1) or (a2), together with Lemma 5.2. Fix a 2-cycle z* * in X such that 0 6= [z] 2 H2(X). We can assume that the coefficient in z of the si* *mplex (K1; K2; K3) is nonzero (otherwise compose with the action of some appropriate * *element of G). Let z* be the corresponding 2-cycle in the barycentric subdivision X* of* * X. Let f : X* ! N ((F0)(K) ) be the G-equivariant simplicial map which sends eac* *h ver- tex in X* to its isotropy subgroup. Thus f(gKi) = {gKig-1}, f(gKij) = {gKijg-1}* *, and f(gK) = {gKg-1}. By conditions (d) and (e), and since neither K2 nor K3 is conj* *ugate to K1, the only simplex in X* which is sent to {K1; K12; K} is (K1; K12; K), an* *d this sim- plex has nonzero coefficient in the 2-cycle z*. Hence {K1; K12; K} has nonzero * *coefficient in the 2-cycle f(z*). By (b) and (c), {K1; K12; K} is maximal in N ((F0)(K) ) (* *not in the boundary of any 3-simplex), and hence [f(z*)] 6= 0 in H2(N ((F0)(K) )) = H2(N (* *F(K)_)) (Lemma 1.4). And thus (G; F) =2U2 by Proposition 1.9(a ) d). * *|__| 5b. Detecting_nonzero_elements_in_H1(X[n]=G)_ Recall that for any n and F, F[n] F denotes the subfamily of all subgroups in* * F of order a multiple of n. We first show, for (G; F) 2 U2, that the connected compo* *nents of the orbit space of N (F[n]) are all acyclic, and then set up some conditions wh* *ich detect elements in their first homology groups. The starting point for all of this is * *the following result, a consequence of Smith theory. Proposition 5.5. If X is any finite dimensional acyclic G-complex, then X=G is * *also acyclic. If f : X ! Y is any equivariant map between finite dimensional G-compl* *exes which induces an isomophism H*(X; Z) ~=H*(Y ; Z), then f=G induces an isomorphi* *sm H*(X=G; Z) ~=H*(Y=G; Z). Proof.The first statement is shown, for example, in [Br, Theorem III.7.12]. The* * second statement follows from the first, since f induces an isomorphism in integral ho* *mology if and only if its mapping cone Cf is acyclic, and similarly for f=G. (Note tha* *t_Cf=G ~= (Cf)=G.) |__| The following result is similar to one used in [O3 ], but formulated here for* * acyclic rather than Fp-acyclic spaces. Proposition 5.6. Fix a prime p, and let X be a finite dimensional acyclic G-com* *plex with the property that XP is acyclic for all p-subgroups P G. Then for any (nonempty) family P of p-subgroups of G, XP =G is acyclic. Proof.We assume that any p-group which contains an element of P also lies in P * *(if not just add these groups to the family). For the purposes of this proof, we de* *fine, for any p-subgroup P G, [ [ XPs= XQ and X(P)s= G.XPs= X(Q): Q%P Q%P Q a p-subgr. Q a p-subgr. We first claim that for any P 2 P, the inclusion of XP into X(P)induces an isom* *orphism of homology groups ~= (P) (P) H*(XP =N(P ); XPs=N(P )) -----! H*(X =G; Xs =G): (1) Bob Oliver and Yoav Segev * * 31 In fact, the inclusion induces an isomorphism ~= (P) (P) C*() : C*(XP =N(P ); XPs=N(P )) -----! C*(X =G; Xs =G) between the cellular chain complexes of these pairs. The surjectivity of C*() i* *s clear, since any open cell oe X(P)rX(P)slies in the G-orbit of some oe XP rXPs. To s* *ee its injectivity, fix open cells oe; a(oe) XP rXPsin the same G-orbit (a 2 G). Then* * P is a Sylow p-subgroup of the isotropy subgroups Goeand Ga(oe)= aGoea-1, so P and a-1* *P a are both Sylow p-subgroups of Goe, and hence a-1P a = gP g-1 for some g 2 Goe. * * It follows that ag 2 NG(P ), and thus that oe and a(oe) = ag(oe) lie in the same N* *(P )-orbit. This proves the injectivity of C*(); and finishes the proof that (1) is an isom* *orphism. Now set ff = max {a 0 | pa|[G:P ]; some P 2 P }: The proposition will be proven by induction on ff. If ff = 0, then for any Syl* *ow p- subgroup P of G, X(P)= XP and X(P)s= XPs= ;; and so H*(XP =G) ~=H*(X(P)=G) ~=H*(XP =N(P )) by (1). Also, XP =N(P ) is acyclic by Proposition 5.5 (since XP is acyclic by a* *ssumption); and thus XP =G is acyclic. Now assume that ff > 0. Let P0 P be the subfamily of all P such that pff|-|[* *G:P ]. Then XP0=G is acyclic by the induction hypothesis, and it remains to show that H*(XP =G; XP0=G) = 0. Let P1; : :;:Pk be conjugacy class representatives for th* *e sub- groups in Pr P0, and set Pi= P0 [ (Pi). Then by excision, Mk Mk H*(XP =G; XP0=G) ~= H*(XPi=G; XP0=G) ~= H*(X(Pi)=G; X(Pi)s=G): i=1 i=1 It thus remains to show that H*(X(P)=G; X(P)s=G) = 0 for each P = Pi. By (1), t* *his means showing that H*(XP =N(P ); XPs=N(P )) = 0. But XPs=N(P ) is acyclic by t* *he induction hypothesis again, and XP =N(P ) is acyclic by Proposition 5.5 (since * *XP_is acyclic by assumption). |* *__| Proposition 5.6 will be applied in particular to get information about the sp* *aces X[n]=G and N (F[n])=G. Corollary 5.7. Let F be any separating family for G, and let F0 F be a subfami* *ly which contains Fc. Let X be a finite dimensional H-universal (G; F)-complex. Th* *en for any subfamily H of F, H*(XH =G) ~=H*(FH =G) ~=H*((F0)H =G): In particular, H*(Xs=G) ~= H*(N (F>1)=G) ~= H*(N ((F0)>1)=G); and H*(X[n]=G) ~= H*(N (F[n])=G) ~=H*(N ((F0)[n])=G) for all n > 1. And for any prime power q, N (F[q])=G is acyclic. Proof.By Proposition 1.3, for any H F, there is a G-map f : X ! N (F) which restricts to a homology equivalence fH : XH ! N (FH ). By Lemma 1.4, the incl* *usion N ((F0)H ) N (FH ) is a homotopy equivalence. So by Proposition 5.5, these m* *aps in- duce homology equivalences in the orbit spaces. The isomorphisms involving H*(X* *s=G) 32 Fixed point free actions on acyclic 2-complexes and H*(X[n]=G) now follow from the case where H = F>1 or H = F[n]. In particula* *r, * * __ X[q]=G is acyclic by Proposition 5.6 (and since X exists by Proposition 1.8). * * |__| The importance of the families F[n]comes from the following lemma. Note that * *for a family F of subgroups of G and a group A of automorphisms of G, the orbit spa* *ce N (F)=A need not be a simplicial complex: there could, for example, be two edg* *es of N (F) not in the same A-orbit, but whose endpoints are identified pairwise. * * But N (F)=A does always have the structure of a CW complex in a natural way (cf. Le* *mma A.5). Lemma 5.8. Let F be a separating family of subgroups of G such that (G; F) 2 * *U2, and let F0 F be any subfamily which contains Fc. Then_for all n > 1, N ((F0)[n* *])=G is homologically 0-dimensional. More generally,_if G Aut(G) is any subgroup_wh* *ich contains Inn(G), and such that F and F0 are G -invariant, then N ((F0)[n])=G i* *s homo- logically 0-dimensional for all n > 1. Proof.Let X be any 2-dimensional H-universal (G; F0)-complex (X exists by Propo- sition 1.8). Then X=G is Z-acyclic by Proposition 5.5. If n = pk where p is p* *rime, then X[n]=G is acyclic by Proposition 5.6. If n is not a prime power,Twrite n =* * q1. .q.k, where the qi are prime powers for distinct primes. Then X[n]=G = ki=1X[qi]=G * *is an intersection of acyclic subspaces of X=G; and hence is homologically 0-dimensio* *nal by Lemma 1.6 again. Thus, N ((F0)[n])=G is also homologically 0-dimensional by Corollary 5.7, and* * its connected components are_all acyclic. The last statement_now follows from Propo* *sition * * __ 5.5, since N ((F0)[n])=G is the orbit space of the G= Inn(G)-action on N ((F0)[* *n])=G. |__| We end this subsection_with an application of Lemma 5.8: one situation in whi* *ch we can show that N (F[n])=G is not homologically 0-dimensional, and thus that (G; * *F) =2U2. The argument is based on the following observation: given a 1-cycle OE in a sim* *plicial complex K which involves at least one "free" edge (an edge with no higher dimen* *sional simplices attached), then 0 6= [OE] 2 H1(K). Here, "simplicial complex" is used* * in the more general sense, where there can be two or more n-simplices (n 1) having th* *e same set of vertices. When working with the orbit space N (F)=G, we will let [H] denote the vertex * *cor- responding to a conjugacy class (H) F. More generally, for any chain H0 $ H1 $ . .$.Hn of subgroups in F, [H0; H1; : :;:Hn] will denote the corresponding n-si* *mplex in N (F)=G. Proposition 5.9. Let F be a separating family of subgroups of G. Assume that th* *ere is a maximal subgroup M 2 F, and a pair of maximal subgroups K; K0 M which are not conjugate in M, but are conjugate in G. Then_(G; F) =2U2. More generally,* * the same conclusion_holds if there is a subgroup G Aut(G) containing Inn(G),_such * *that F is G -invariant, and such that K and K0 are in the same orbit of G , but not * *in the same orbit of the action of the stabilizer of M. __ Proof.Set n = |K|. Then F[n]=G contains (at least) two edges which connect the * *vertices [K] and [M]._The maximality properties guarantee that the resulting loop is non* *zero in * *__ H1(F[n]=G ). So (G; F) =2U2 by Lemma 5.8. |* *__| Bob Oliver and Yoav Segev * * 33 5c. Acyclicity_of_N_(F>1)_ We now find conditions for showing that N (F>1) is acyclic, under the assumpt* *ion that (G; F) 2 U2. This can then be combined with results in Section 5d to obta* *in contradictions. We first note the following equivalent conditions on F. Lemma 5.10. Fix a separating family F of subgroups of G, and assume that (G; * *F) 2 U2. Then the following are equivalent: (a) N (F>1)=G is connected and H1(N (F>1)=G) = 0. (b) N (F>1) is acyclic. (c) N (F>1)=G is acyclic. Proof.For any 2-dimensional H-universal (G; F)-complex X, H*(Xs) ~= H*(N (F>1)) by Proposition 1.3, and H*(Xs=G) ~=H*(N (F>1)=G) by Corollary 5.7. So it suffic* *es to show the equivalence of the above three conditions after replacing N (F>1) by X* *s. Since X=G is acyclic (Proposition 5.5), Xs and Xs=G are homologically 1-dimen* *sional by Lemma 1.6. Thus, (a) is equivalent to (c). Also, (b) implies (c) by Proposit* *ion 5.5 again; and it remains to show that (c) implies (b). If Xs=G is acyclic, then in particular it has Euler characteristic one. Hence* * by Lemma 2.2, 1 - O(Xs) = O(X) - O(Xs) = |G|. O(X=G) - O(Xs=G) = |G|(1 - 1) = 0; and so O(Xs) = 1. Since G acts transitively on the connected components of Xs (* *Xs=G being connected), all components of Xs have the same Euler characteristic, and * *so Xs must be connected. And since Xs is also homologically 1-dimensional, this shows* *_that_ Xs is acyclic. |* *__| The next proposition provides a tool for showing that condition (a) in Lemma * *5.10 holds. Proposition 5.11. Assume G has even order, and let F be a separating family for* * G. Assume, for each member M 2 Fmax of even order and each element x 2 M of odd prime order, that either (1a) |NM ()| is even; or (1b) there is an element y 2 M of odd prime order such that |NG()\ NG()* *| and |NM ()| are both even. Let (M1); :::; (Mk) be the conjugacy classes of odd order subgroups in Fmax. F* *or 1 i k, let F0ibe the set of all subgroups of Miwhich are contained in members of* * Fmax of even order or in subgroups conjugate to Mj for j < i; and assume that (2) the image of N ((F0i)>1) in N (F>1)=G is connected and nonempty for each * *i. Then N (F>1)=G is connected and H1(N (F>1)=G) = 0. Proof.For any x 2 H G, we write NH (x) def=NH (), for short. For each i = 0* *; :::; k, let Fi be the family of all subgroups in F contained in even order members of F* *max, or in subgroups conjugate to Mj for j i; and set Xi = N ((Fi)>1)=G and X = Xk. In particular, Fk = F, and F0 is the set of all subgroups in F which are contai* *ned in 34 Fixed point free actions on acyclic 2-complexes members of Fmax of even order (k = 0 if all members of Fmax have even order). * *Set Y = N (F[2])=G X0. By Corollary 5.7, Y is connected and H1(Y ) = 0. Then X0 is connected, since each vertex of X0 is joined by an edge to a vertex of Y . And * *for each i 1, each vertex of Xi not in Xi-1 is joined to [Mi], which in turn is connect* *ed to Xi-1via a vertex in the nonempty set F0i. This shows that the Xiare all connect* *ed. In particular, X is connected, and it remains to show that H1(X) = 0. We first set up some notation for elements of H1(X). The homology class of a * *loop will be denoted [H0; H1; : :;:Hn], where (H0) = (Hn), and each Hicontains or is cont* *ained in Hi+1. Note that by specifying subgroups rather than just conjugacy classes, we * *eliminate all ambiguity as to which edge between two vertices is meant (recall that there* * can be more than one edge connecting a pair of vertices of X). Finally, to simplify th* *e notation, we will sometimes replace a cyclic group Hi= by xi in this notation. Step 1 We first show that H1(X0) maps trivially to H1(X). Whenever [H0; H1; : :* *;:Hn] is a path in X with endpoints in Y , we write [H0; H1; : :;:Hn]Y 2 H1(X) to den* *ote the homology class of the 1-cycle [H0; : :;:Hn] - OE for any path OE from [H0] to [* *Hn] in Y . This is well defined since Y is connected and H1(Y ) = 0. Fix a loop in X0; we can assume that it alternates "peaks" and "valleys" (ver* *tices corresponding to larger or smaller subgroups); and furthermore that each peak i* *s max- imal in F (hence of even order) and each valley is minimal (i.e., of prime orde* *r). The loop thus splits into a sum of elements [M; x; M0]Y , where M and M0 are maxima* *l of even order, and where |x| is prime. If |x| = 2, then [M; x; M0]Y 2 Im(H1(Y )) =* * 0; so we can assume that x has odd prime order. In either of cases (1a) or (1b) above, NG(x) has even order. Choose a maximal* * sub- group Mx 2 F[2]which contains the extension of by a Sylow 2-subgroup of NG(* *x)= (this extension is solvable and hence in F[2]). Then [M; x; M0]Y = [M; x; Mx]* *Y + [Mx; x; M0]Y , and we are reduced to showing that [M; x; Mx]Y = 0 in H1(X). If |NM (x)| is even, let H M be any subgroup which contains with index 2. Then H is conjugate in NG(x) to some H0 Mx (by choice of Mx); and so [M; x; Mx]Y = [M; H; x; H0; Mx]Y = [M; H]Y + [H0; Mx]Y (the last equality holds* * be- cause [H; x] = [H0; x]). But these edges lie in Y = N (F[2])=G, and so [M; x; M* *x]Y = 0. Thus, [M; x; Mx]Y = 0 whenever x 2 M satisfies condition (1a). Now assume that x 2 M satisfies condition (1b), and fix y 2 M as in (1b). Fix subgroups My; Mxy2 Fmax of even order, such that My contains the extension of <* *y> by a Sylow 2-subgroup of NG(y)=, and Mxy contains the extension of by a * *Sylow 2-subgroup of NG(x)\ NG(y) (this last extension must lie in F since M 2* * F and F is separating). Consider the following diagram: M x ? ? Mx --- ---! --- ---! My ? ? y Mxy Bob Oliver and Yoav Segev * * 35 By construction, condition (1a) is satisfied by each of the pairs x 2 Mxy, y 2 * *Mxy and y 2 M, and so [M; ; y; My]Y = 0 = [Mxy; ; y; My]Y = [Mxy; ; x; Mx]Y* * : And hence [M; x; Mx]Y = [M; ; x; Mx]Y = 0. Step 2 We now prove inductively, for i 1, that H1(Xi) has finite image in H1(X) if H1(Xi-1) does. Fix a loop in Xi. We can again assume that it alternates "p* *eaks" and "valleys"; and that each peak is either equal [Mi] or lies in Xi-1. If any* * of the valleys is a vertex [H] =2Xi-1, then it must be connected on both sides to [Mi]* * (but possibly by different edges). This forms a loop (two edges each connectingf[H]i* *to [Mi]) whose homology class lies in the image of H1(F[p])=G for any prime pfi|H|, and * *this group vanishes by Corollary 5.7. We are thus reduced to looking at 1-cycles of * *the form z = OE - [H; Mi; H0], where H; H02 F0iand OE is a path in Xi-1connecting [H] an* *d [H0]. And since the image of N ((F0i)>1) in X is connected by (2), the path [H; Mi; H* *0] is homotopic to a path in Xi-1(and hence [z] is in the image of H1(Xi-1)), modulo * *loops_ of the form [K; Mi; K0] for G-conjugate subgroups K; K02 F0i. * * |__| The following proposition shows that in certain cases, one can replace F by a* * different separating family without changing the homology of N (F>1) or of N (F>1)=G. Not* *e, in its statement and proof, that any finite group G contains a (unique) maximal no* *rmal perfect subgroup L C G: the last term in the derived series of G. This normal s* *ubgroup is also characterized by the properties that L is perfect and G=L is solvable. Proposition 5.12. Let F0 $ F be two separating families in G, and let H F be a* *ny subfamily. Assume that one of the following two conditions holds: either (a) for each perfect subgroup L 2 Fr F0, there is a solvable subgroup N C CG(* *L) with N 2 H; or (b) the maximal normal perfect subgroup Lmax C G is simple, and CG(L) 2 H for each perfect subgroup L 6= Lmax in S(G)r F0. Then the inclusion of N (F0H ) into N (FH ) is a homotopy equivalence, and H*(N (F0H )=G) ~=H*(N (FH )=G): Proof.Note that the set of perfect subgroups in Fr F0 is nonempty. Since for a* *ny H 2 Fr F0 with maximal normal perfect subgroup L C H, L 2 Fr F0 since H=L is solvable. We first check that condition (b) implies condition (a). Fix any perfect sub* *group L 2 Fr F0, and let L0 L be the maximal normal perfect subgroup of L.CG(L). Then CG(L0) CG(L), so L0.CG(L0) L.CG(L), and CG(L0) is solvable since (L0.CG(L0))=* *L0 is solvable and L0\ CG(L0) = Z(L0) is abelian. Also, CG(L) normalizes L0, and * *so CG(L0) C CG(L). If (b) holds, then either L = L0 or L0 is not simple (since L C* * L0); and in either case L0 6= Lmax and so CG(L0) 2 H. Condition (a) thus applies, w* *ith N = CG(L0). Now assume that condition (a) holds. Fix a conjugacy class L of maximal perf* *ect subgroups in Fr F0. Set F00= Fr (FL ): the family of subgroups in F which do not contain any subgroup in L. This is a separating family (if H=K is solvable* * and H L 2 L then K L); and we can assume inductively that the inclusion of N (F0>* *1) 36 Fixed point free actions on acyclic 2-complexes into N (F00>1) is a homotopy equivalence. So upon setting F0 = F00, we are red* *uced to the case where Fr F0 contains a single conjugacy class L of perfect subgroup* *s, and where F0 is the set of subgroups in F which do not contain any subgroup in L. For each L 2 L, let KL be the set of all subgroups H N(L) such that HL=L is solvable, and let K0Lbe the set of all H 2 KL such that L 6 H. Then KL F (HL=L solvable implies HL 2 F and hence H 2 F), and K0L= KL \ F0. By assumption, there is a solvable normal subgroup N C CG(L) with N 2 H. Upon replacing N by the subgroup generated by its conjugates in N(L) (still solvable since it is ge* *nerated by solvable normal subgroups of CG(L)), we can assume that N C N(L) (and N 2 FH ). Then HN 2 KL for all H 2 KL (HNL=L is solvable if HL=L is since HL=L normalizes NL=L and NL=L is solvable). Also, HN 2 K0Lfor all H 2 K0L: since for H 2 KL, H=(H \ L) ~=HL=L is solvable, so HN=(H \ L) is solvable (since N is solvable and centralizes H\L), and thus HN contains L if and only if H does. The nerves N ((* *KL)H ) and N ((K0L)H ) are thus contractible by Lemma 0.4(b). For each subgroup H 2 Fr F0, there is a unique L 2 L contained in H: the subg* *roups in L are maximal among perfect subgroups in Fr F0, and hence L must be the last term in the derived sequence for H. Thus, L C H and H=L is solvable; and L is t* *he unique element of L for which H 2 KL r K0L. In other words, N (FH ) is the un* *ion of N (F0H ) with the contractible complexes N ((KL)H ) for L 2 L, any two of * *the complexes N ((KL)H ) and N ((KL0)H ) have intersection contained in N (F0H )* *, and N (F0H ) \ N ((KL)H ) = N ((K0L)H ) is also contractible for each L. The inc* *lusion of N (F0H ) into N (FH ) is thus a homotopy equivalence; and hence H*(N (F0H )=* *G)_~= H*(N (FH )=G) by Proposition 5.5. |* *__| 5d. Connectivity_of_links_at_vertices_ In Section 5c, conditions were found on a separating family F which imply tha* *t if (G; F) 2 U2, then N (F>1) is acyclic, and hence there is a 2-dimensional H-univ* *ersal (G; F)-complex with no free orbits. The results of this section amount to showi* *ng that if there is such an action, then the links at all of its vertices must be conne* *cted. This result, and its proof, are closely related to [S2, Theorem 2.8]. Proposition 5.13. Let F be a nonempty family of subgroups of G, such that G =2F. Let Fmax be the set of maximal members of F. Assume that (a) each member of Fmax is self-normalizing; (b) each member of Fr Fmax is contained in at least two members of Fmax; and (c) N (F) is connected and H1(N (F)) = 0. Then for each M 2 Fmax, N (F1(M) = N (F1) = N {H 2 F | 1 6= H $ M} : Proposition 5.14. Fix a separating family F for G. Let F0 F be any subfamily which contains Fc, and such that each nonmaximal subgroup in F0 is contained in* * two or more maximal subgroups. Assume that F satisfies the following two conditions: (a) N (F>1)=G is connected and H1(N (F>1)=G) = 0. (b) There is a maximal subgroup M 2 F such that Lk(F0)>1(M) is not connected. Then (G; F) =2U2. Proof.Assume that (G; F) 2 U2. Then by (a) and Lemma 5.10, N (F>1) is acyclic. * *So Proposition 5.13, applied to the family (F0)>1, implies that Lk(F0)>1(M) = N ((* *F0)1) is connected for all maximal subgroups M 2 F, and this contradicts point (b). (* *Recall_ that all maximal subgroups in F are self-normalizing by Lemma 1.1.) * * |__| 38 Fixed point free actions on acyclic 2-complexes 6.Simple groups of Lie rank one We now focus attention on the simple groups of Lie type and Lie rank one. Th* *ere are four families of such groups: the two dimensional projective special linear* * groups L2(q), the three dimensional projective special unitary groups U3(q), the Suzuk* *i groups Sz(22k+1), and the Ree groups Ree(32k+1) = 2G2(22k+1). We refer to Appendix D * *for more detail on these groups, and on the classification of finite groups of Lie * *type in general. We first show that the only 2-dimensional actions which involve the simple gr* *oups L2(q) or Sz(q) are the ones constructed in Section 3. This will be done in a se* *ries of three lemmas, after which the results will be summarized in Proposition 6.4. Lemma 6.1. Assume that G = L2(q) or P GL2(q), where q = pk and p is an odd pr* *ime. Let F be a separating family for G which contains no nonsolvable subgroups L2(q* *0) or P GL2(q0) for q0 a smaller power of p. Assume also that F 6= SLV if G = L2(q) * *and q 3 (mod 8). Then (G; F) =2U2. Proof.We refer to the description of maximal subgroups of G in Proposition 3.3.* * Note that if G = L2(q) and q 3 (mod 8), then F must contain a subgroup isomorphic to A5 _ the only nonsolvable subgroups of G not isomorphic to L2(q0) or P GL2(q0) * *for q0 a smaller power of p. In particular, q 1 (mod 5) in this case. Case 1: Assume first that p = 3. If k is odd, then q 3 (mod 8) and q 2 (mod 5); and so G 6~= L2(q) by the above remarks. Thus, either G = L2(3k) for k eve* *n, or G = P GL2(3k). Set K1 = P GL2(3) ~=4 (the subgroup of matrices with entries in F3), let K2 b* *e the subgroup of upper triangular matrices (K2 ~=Fqo C(q-1)=2or Fqo Cq-1), and let K* *3 be the subgroup of monomial matrices (K3 ~=Dq-1 or D2(q-1)). Set Kij= Ki\ Kj and K = K1 \ K2 \ K3. Then K12~= D6, K13~= C22, K23~= C(q-1)=2or Cq-1, and K ~=C2. Since K1 is a maximal subgroup in F (see the list of subgroups in Proposition 3* *.3), Proposition 5.4 now applies (using condition (a1), or (a2) if G = L2(9)) to sho* *w that (G; F) =2U2. Case 2: Now assume that p 5. By Proposition 3.3, A4 is a maximal subgroup of G only if G = L2(q) and q 3 (mod 8), in which case (as noted above) F must conta* *in subgroups isomorphic to A5. And since there is only one conjugacy class of A4 * * G (Proposition 3.3 again), each such subgroup must be contained in some A5 2 F. Thus, no maximal subgroup of F is isomorphic to A4. From the lists of maximal subgroups in Proposition 3.3, we now see that each maximal subgroup in F is iso* *morphic to one of the groups Fqo C(q-1)=2or Fqo Cq-1(triangular matrices); Dq-1or D2(q-* *1); Dq+1 or D2(q+1); or 4 or A5. Also, by hypothesis, if p = 5, then A5 ~=L2(5) is not i* *n F. Let M1 G be the (maximal) subgroup of upper triangular matrices, and let T * *M1 be the subgroup of diagonal matrices. From the above list (and since p > 3) we * *see that M1 and its conjugates are the only maximalfsubgroupsiin F of order a multiple o* *f p. Furthermore, for any subgroup H 2 F with pfi|H|, H leaves invariant a unique li* *ne in (Fq)2, and hence is contained in a unique subgroup conjugate to M1 (and thus a * *unique maximal subgroup in F). Also, each nontrivial subgroup H M1 of order prime to p is contained in a unique subgroup conjugate to T (i.e., CM1(H)). Bob Oliver and Yoav Segev * * 39 We first check that N (F>1)=G is connected and H1(N (F>1)=G) = 0, using Propo* *si- tion 5.11. From the above list of maximal subgroups in F (and since A4 is not a* *mong them), we see that for each maximal subgroup M 2 F of even order, and each x 2 M of odd prime order, NM () has even order. Thus, condition (1a) in Propositio* *n 5.11 holds. Also, the only maximal subgroups in G of odd order are those conjugate * *to M1 ~=Fqo C(q-1)=2, when G = L2(q) and q 3 (mod 4). Let F01be the set of subgro* *ups of M1 which are contained in maximal subgroups in other conjugacy classes; by t* *he above remarks each H 2 F01is conjugate to a subgroup of T . The image of N ((F0* *1)>1) in N (F>1)=G is thus connected, and so condition (2) in Proposition 5.11 holds.* * This finishes the proof that N (F>1)=G is connected and H1(N (F>1)=G) = 0. Now let F0 F be the subfamily consisting of all maximal subgroups in F, toge* *ther with all subgroups in F contained in two or more maximal subgroups. We have seen that each proper subgroup of M1 contained in F0 is contained in a unique subgro* *up conjugate to T . In other words, Lk(F0)>1(M) = N ((F0)1) is not connected: i* *t has one connected component for each subgroup of M1 conjugate to T . So Proposition 5.1* *4_now_ applies to show that (G; F) =2U2. * *|__| In each of the next two lemmas, we deal simultaneously with simple groups L =* * L2(q) and Sz(q), where q = pk and p is prime (p = 2 if L = Sz(q)). It will be conveni* *ent to fix subgroups S; T; B; N L of each of these groups, according to the following tab* *le: ______________________________________________________ || L|| L2(q) | Sz(q) || ||__||___________________________|____________________ || || S|| { 1a | a 2 Fq} ~=Fq | S(q; ) || ||__||_________01________________|___________________ || || T|| {diag(; -1) | 2 (Fq)*}={I}| {M() | 2 (Fq)*} || ||__||___________________________|___________________ || || B|| So T | M(q; ) = S(q; )o T || ||__||___________________________|___________________ || || N|| N(T ) = | N(T ) = || ||__||___________________-10_____|____________________|| Table 4 When L = Sz(q), we are using the notation in Proposition 3.6 (where Sz(q) is re* *garded as a subgroup of GL4(q)). All of these subgroups are invariant under the actio* *n of Aut(Fq). In both cases, S is a Sylow p-subgroup, B = N(S) is a Borel subgroup, * *T is cyclic (of order q- 1 or (q- 1)=2), and N is dihedral. Lemma 6.2. Assume that G = L is one of the simple groups L2(q) or Sz(q), where q = pk and p is prime (p = 2 in the second case). Let F be any separating famil* *y for G which contains a nonsolvable subgroup isomorphic to L2(q0) or Sz(q0), where q0 * *= pk0 (and k0|k). Then (G; F) =2U2. Proof.Assume that q0 = pk0 is chosen so that F contains a maximal subgroup iso- morphic to G0 = L2(q0), P GL2(q0), or Sz(q0). Thus, G0 is the subgroup of all * *ma- trices in G with entries in Fq0. (More precisely, if G = L2(q) P GL2(q), then G0 = L2(q) \ P GL2(q0).) By Proposition 3.3 or 3.6, if G0 ~=M 2 F, then there * *is an automorphism oe 2 Aut(G) such that oe(M) = G0. Thus, upon replacing F by oe(* *F), we can assume that G0 2 F. We now apply Proposition 5.4, with the subgroups K1 = G0, K2 = B, and K3 = N (as in Table 4). Then K12 = B0, K13 = N0, K23 = T , and K = K1 \ K2 \ K3 = T0. 40 Fixed point free actions on acyclic 2-complexes Condition (b) of 5.4 holds by assumption (K1 = G0 is maximal in F). Conditions * *(d) and (e) are clear: NG(G0; B0; T0) = T0, and the triples (K1; B0; T0) and (K1; N* *0; T0) are not G-conjugate. We next consider condition (c). Clearly, K12= B0 is a maximal subgroup of K1 * *= G0. If G = L2(q), then K = T0 is maximal in K12 = B0. And if G = Sz(q), then T0 is maximal among critical subgroups of B0. (There is one subgroup T0 $ R $ B0, whe* *re R = Z(S(q0; )).T0 ~=Fq0oCq0-1. But using Proposition 3.6(b), it is easy to chec* *k that every maximal subgroup of G which contains R also contains B0. So by Lemma 3.1(* *a), R is not critical.) It remains to check that inequality (a1) or (a2) holds. From the above descri* *ption of the groups, we see that ( q0 if L = L2(q) q-1 [K12:K] = [B0:T0] = [K13:K] = 2; [K23:K] = ffl.____q* *;-1 (q0)2 if L = Sz(q), 0 where ffl = 1_2if G = L2(q),Pp is odd, and 2k0|k (so G0 = P GL2(q0)), and ffl =* * 1 otherwise. Inequality (a1) now holds ( i 2 is a power of 2. Then there is* * no 2-dimensional G-complex without fixed points. Proof.Assume otherwise: let q be such that U3(q) is the smallest counterexample* *, and let F be a separating family of subgroups of G such that (G; F) 2 U2. Set d = (3; q+ 1). Then 2-q+1 |G| = 1_dq3(q2 - 1)(q3 + 1) = q3.(q - 1).(q + 1)2.(q_____d) (1) (cf. [Ca , Theorem 14.3.2], who writes U3(q) = 2A2(q2)). Here, the factors in t* *he second 3+1 formula are pairwise relatively prime. (Note that 3|(q2 - q + 1) = q___q+1if a* *nd only if 3+1 2 3|(q+ 1), and that q___q+1cannot be divisible by 3 .) Let be the Frobenius automorphism of order 2 on Fq2; and write x = (x) = xq* * for any x. The following list of maximal subgroups of G can be found in [Ha , p. 158] o* *r in [GLS , Theorem 6.5.3(a,b,c,g)]. Note also the thesis of Peter Kleidman [Kl1, x5* *], where maximal subgroups are listed for classical groups of low rank, and a general pr* *ocedure for determining them is described. (M1): M1 ~=[q3]o C(q2-1)=d; the stabilizers of isotropic lines (generated by v* * with (v; v) = 0); the Borel subgroups of G. We choose M1 to be the stabilizer of , or e* *quivalently the group of upper triangular matrices with respect to the basis {v1; v2; v3}. (M2): M2 ~=GU2(q)=Cd ~=C(q+1)=dxL2(q); the stabilizers of anisotropic lines (g* *enerated by v with (v; v) 6= 0). We choose M2 to be the subgroup of matrices (aij) (wi* *th respect to either of the above bases) for which a22is the only nonzero entry in the s* *econd row or column. 44 Fixed point free actions on acyclic 2-complexes (M3): M3 ~=[(Cq+1)2o 3]=Cd; the stabilizer of (the union of) three pairwise or* *thogonal lines. We choose M3 to be the group of monomial matrices with respect to the orthonormal basis {e1; e2; e3}. (Mq04), if q = qb0and b is an odd prime: Mq04= N(U3(q0)), isomorphic to U3(q0)* * (if (b; d) = 1) or P GU3(q0) (if b = d = 3). There are (b; d) conjugacy classes * *of such subgroups (all conjugate in P GU3(q)). (M5): M5 ~=C(q2-q+1)=doC3. Consider the hermitian form <-; -> on Fq6(viewed as* * a 3 vector space over Fq2) defined by = Tr(xyq ), where Tr: Fq6! Fq2is the* * trace 2 * map. Let oe 2 Aut (Fq6) be the automorphism oe(x) = xq , let H (Fq6) be the subgroup of order q3 + 1, and set M = Ho (Fq6)* o Aut(Fq6): Then M preserves <-; ->, and M5 is the intersection of U3(q) with the image o* *f M 2 in P GU3(q). In particular, C3 acts on C(q2-q+1)via x ! xq . We can assume inductively that none of the groups Mq04= N(U3(q0)), for q0 > 2* *, can act on an acyclic 2-complex without fixed points. So they must all be contained* * in F. Also, by Proposition 6.4, if M2 =2F, then the only subgroups of M2 (and its con* *jugates) which are in F are solvable subgroups. So either F = MAX , the family of all * *proper subgroups of G, or F = F0, the family of all subgroups whose intersection with * *any subgroup in (M2) is solvable _ and this latter only when k is prime or a power * *of 2. We first show that N (F>1)=G is connected and H1(N (F>1)=G) = 0, using Propos* *ition 5.11. Since every perfect subgroup in MAX r F0 is of the form L2(2k0) where * *1 < k0|k and has nontrivial centralizer, Proposition 5.12 applies, with H = MAX >* *1 (and using condition (b)) to show that H*(N ((F0)>1)=G) ~=H*(N (MAX >1)=G). So we * *can assume that F = MAX . The even order maximal subgroups of G are those conjugate to M1, M2, M3, or M* *q04. If M = M2, M3, or Mq04and x 2 M is of odd prime order,fthenione easily sees that NM () has even order. Also, if x 2 M1 and |x|fi(q + 1)=d, then CM (x) has ev* *en order: if M1 is the subgroup of upper triangular matrices with respect to the basis {v* *1; v2; v3}, thenixjis conjugate to a diagonal matrix diag(; -2; ) and is centralized by the* * element 100110 0 01 . Thus, condition (1a) of Proposition 5.11 holds in all of these cases. Now let x 2 M1 be of prime order dividing p - 1. We check that condition (1b) of Proposition 5.11 holds. Let y 2 CM (x) ~=C(q2-1)=dbe any element of prime o* *rder dividing (q+1)=d. We have just seen that NM () has even order, and NG()\N* *G() also has even order since NM () ~= M2 ~=C(q+1)=dxL2(q). Thus, condition (1b* *) of Proposition 5.11 holds in this case. It remains to check condition (2) of 5.11. Let F1 be the set of all subgroup* *s of M5 ~=C(q2-q+1)=doC3 which are also contained in even order maximal subgroups. By inspection, F1 contains subgroups of order 3, and all maximal subgroups in F1 a* *re of the form Cao C3 ( Mq04) for some a. So the image of N ((F1)>1) in N (F>1)=G is none* *mpty and connected. Proposition 5.11 thus applies to show that N (F>1)=G is connecte* *d and H1(N (F>1)=G) = 0. This shows that condition (a) in Proposition 5.14 holds, and it remains to ch* *eck condition (b). Set M = M5 ~=C(q2-q+1)=doC3. Let Fc F be the subfamily consisti* *ng Bob Oliver and Yoav Segev * * 45 6-1)(q-1) of all critical subgroups in F. Fix a prime p|(q2 - q + 1) = (q_______(q3-1)(* *q2-1)such that p|-|(q60- 1) when q0 is a smaller power of 2 (such a prime existsfbyiZsigmondy'* *s theorem [HB2 , Theorem IX.8.3]). Then for any proper subgroup H M with pfi|H|, M is the unique maximal subgroup of G which contains H, and so H =2Fc (Lemma 3.1(a)). Let T C M be the subgroup of order (q2 - q + 1)=dp; then M=T ~=Cpo C3. And C3 is not 2 2 2 normal in M=T : since C3 acts on Cp via (x 7! xq ), and (q - 1; (q - q + 1)=d* *) = 1. By Proposition 5.14, we will be done upon showing that the nerve of (Fc)1i* *s not connected. For any 1 6= H T , H is not critical by Lemma 3.1(b): N(H) = M 2 F, and NK (H) % K for all K % H (note that K must be contained in M or in one of t* *he subgroups N(U3(q0))). Thus, any critical subgroup properly contained in M must * *be of the form Ho C3 for H T ; and such subgroups do exist (any subgroup of M maximal among those contained in other maximal subgroups in F is critical). The image o* *f the poset (Fc)1in S(M=T ) thus consists precisely of the subgroups of order 3. S* *ince the continuous image of a connected space must be connected, this shows that N ((Fc* *)1)__ is not connected, and finishes the proof of the proposition. * * |__| We note here that Proposition 6.6 can also be proven using Propositions 5.3 a* *nd 5.4; but this involves considering several different cases, and requires complicated* * arguments that certain subgroups are not critical. We are now ready to consider the Ree groups 2G2(q). Proposition 6.7. When q is any odd power of 3, there is no 2-dimensional 2G2(q)- complex without fixed points. Proof.Set G = 2G2(q), where q = 3k and k is odd; and assume that F is a separat* *ing family for G such that (G; F) 2 U2. We can assume inductively that q is the sma* *llest power of 3 for which this happens. Since 2G2(3) ~=Aut(L2(8)) [Jan], this subgro* *up has no fixed point free action on a Z-acyclic 2-complex by Lemma 6.3. Thus, we must* * have 2G2(3) 2 F. The order of G is given by the formula p__ p__ |G| = q3(q - 1)(q3 + 1) = q3.23.(q-1_2).(q+1_4).(q + 3q + 1).(q - 3q + * *1); (cf. [Ca , Theorem 14.3.2]), where the factors in the last decomposition are p* *airwise relatively prime. The maximal solvable subgroups of G, as listed in [Kl2, Theor* *em C], all lie in the following conjugacy classes: (M1): the Borel subgroups P oCq-1, where |P | = q3 (a Sylow 3-subgroup of G). * *More precisely, P = (Fq)3 with multiplication given by (x1; y1; z1).(x2; y2; z2) = (x1 + x2; y1 + y2 + x1.xoe2; z1 + z2 - x1.y2 + y1.* *x2 - x1.xoe1.x2): p__ 2 Here, xoe= x 3q (so xoe= x3). The action of (Fq)* on P is given by (x; y; z)-1 = (x; oey; 2oez): (See [HB3 , Theorem XI.13.2].) (M2): M2 = CG(C2) ~=C2x L2(q) for any C2 G (M3): M3 = N(C22) = (C22xD(q+1_2))o C3 for any C22 G 46 Fixed point free actions on acyclic 2-complexes (M+4) and (M-4): M4 ~= Cqp__3q+1oC6, where C6 acts via (x 7! xq). (The action* * of C6 is determined by the fact that an element of order 2 or 3 has centralizer * *of order prime to q2- q+ 1.) (Mq05): Mq05~=2G2(q0) whenever q = qp0for some (odd) prime p. By our inductive assumption, 2G2(q0) 2 F for all q0 = 3k0 where k0|k. So all * *of the maximal subgroups must be included in F, except possibly those in (M2). We first show that N (F>1) is connected and that H1(N (F>1)=G) = 0. By Propos* *ition 5.12 (arguing as in the proof of Proposition 6.6), it suffices to do this when * *F = MAX : the family of all proper subgroups of G. We apply Proposition 5.11. From the ab* *ove list, we see that all maximal subgroups of G have even order. If M is maximal * *and x 2 M has odd prime order, then NM (x) has even order, except possibly when M is conjugate to M1 and |x| = 3. And under the above description of P C M1, any x 2* * P of order 3 is of the form x = (0; b; c) for b; c 2 Fq; x is normalized by (-1) * *2 (Fq)* if b = 0 or c = 0; and if b 6= 0 then x = (0; b; c) is conjugate to (0; b; 0). Con* *dition (1a) of Proposition 5.11 thus holds (and condition (2) is empty). It follows that N * *(F>1) is connected and H1(N (F>1)=G) = 0. We have now shown that condition (a) in Proposition 5.14 holds. We claim that condition (b) holds for one of the maximal subgroups M4 ~= Cqp__3q+1oC6; once t* *his has been shown then we can conclude that (G; F) =2U2. By Zsigmondy's theorem [H* *B2 , Theorem IX.8.3], there is a prime p|(q6 - 1) = (36k- 1) such that p|-|(3m - 1) * *for any 3+1 p__ p__ m < 6k. In particular, p|q___q+1= (q + 3q + 1)(q - 3q + 1) _ and thus divides* * the order of M = M+4or M-4_ but does not divide the order of 2G2(q0) for any q0 < q. We claim that the nerve of the poset of proper subgroups of M which are critica* *l in F is not connected. Let T C M be the cyclic subgroup of index 6p, and set H = Im[(Fc)1---! S(M=T )]: From the above list of maximal subgroups, we see that for any proper subgroup H* * $ M of order a multiple of p, H is contained in no other maximal subgroup in F, and* * hence H is not critical (Lemma 3.1(a)). Also, for any 1 6= H T , Lemma 3.1(a) applie* *s (with bH= N(H) = M) to show that H =2Fc. Thus, H contains neither the trivial subgroup nor subgroups of order a multiple of p. Also, H contains the subgroups of orde* *r 6 in M=T , since any subgroup of the form Ho C6 M (for H T ) which is maximal among subgroups of M contained in other maximal subgroups of F must be critical. We h* *ave now shown that H consists of the subgroups of order 6 in M=T ~= Cpo C6, as well* * as possibly the subgroups of order 2 and 3. Since none of these subgroups is norma* *l (C6 acts on Cp via (x 7! xq) and p is prime to (q2 - 1) and to (q3 - 1)), this show* *s that the nerve of H is not connected. And since the continuous image of a connected spac* *e must __ be connected, this shows that N ((Fc)1) also fails to be connected. * * |__| Proposition 6.7 can also be proven using Proposition 5.4 (when F contains cen* *tralizers of involutions), and Proposition 5.12 to reduce the general case to this case. Bob Oliver and Yoav Segev * * 47 7. Sporadic simple groups Aschbacher and Segev proved in [AS ] that no sporadic simple group, with the * *possi- ble exception of the first Janko group J1, can act on a 2-dimensional acyclic c* *omplex without fixed points. In all cases, this was done by applying the four-subgrou* *p crite- rion, presented here in Proposition E.1. Since the arguments in [AS ] use a va* *riety of structures and definitions unfamiliar to non-group-theorists, we now describe h* *ow these results _ as well as the nonexistence of a J1-action _ can be proven using Prop* *osi- tion 5.4 instead. Note however that the arguments presented here, while fairly* * brief to present, are not really more elementary than those given in [AS ]. They depe* *nd on information about maximal subgroups which has been collected together in [Atl] * *and [A2 ], but whose proofs (especially for the ten sporadic groups listed in Table* * 5) are scattered widely throughout the literature. We first repeat some definitions in [A2 , x28]. Fix a finite group G, a subgr* *oup A Aut(G), and an A-invariant subgroup B G. A regular (A; B)-basis for G is a set {Gi| i 2 I} of subgroups containing B which satisfies the following two conditi* *ons: T (1) each subgroup H G containing B is in the A-orbit of GJ def=j2JGj for some unique J I (in particular, B = GI); and (2) for each J; K I, if a(GK ) GJ for some a 2 A, then GK GJ and a(GK ) = a0(GK ) for some a02 NA(GJ). If G has a regular (A; B)-basis of order at least four (for any A and B), then * *by [AS , 6.1] (and using the four-subgroup criterion described in Proposition E.1), (G; * *F) =2U2 for any separating family F which contains the basis. Using Proposition 5.4, th* *is can be shown for bases of order three which satisfy certain additional conditions. Lemma 7.1. Fix a simple group G and a separating family F of subgroups of G. Assume, for some A Inn(G) and some A-invariant subgroup 1 6= K G, that there is a regular (A; K)-basis {Ki| i 2 I}, and indices r; s; t 2 I, such that Kr; K* *s; Kt 2 F and ____1____ 1 1 + _________+ _________ 1: (1) [Krs:Krst] [Krt:Krst] [Kst:Krst] Then (G;fF)i=2U2. In particular, (1) holds if K contains a Sylow p-subgroup for* * any prime pfi|G|. Proof.For simplicity, we write I = {1; 2; : :;:k}, and assume that {r; s; t} = * *{1; 2; 3}. By [A2 , 28.1], {K1; K2; K3} is a regular (NA(K123); K123)-basis; so we can ass* *ume k = 3 and K = K123. It is immediate from the definition of a regular (A; K)-basis tha* *t KJ[{i} is a maximal subgroup of KJ for any J $ I and any i 2 Ir J. We claim that the subgroups K1; K2; K3 satisfy the hypotheses of Proposition * *5.4; it then follows that (G; F) =2U2. We have just checked conditions (b) and (c) (max* *imal- ity of subgroups). Condition (a1) holds by assumption, and condition (e) (the t* *riples (K1; K12; K) and (K1; K13; K) are not G-conjugate) is immediate from the defini* *tion of a regular (A; K)-basis. We next show that the Kican be ordered so that NG(K1; K12; K) = K, thus provi* *ng condition (d). To see this, note first that NG(K) must be A-conjugate (hence e* *qual) 48 Fixed point free actions on acyclic 2-complexes to one of the subgroups Ki, Kij, or K. Also, the Ki are maximal in the simple g* *roup G and hence self-normalizing. If NG(K) = K, then we are done. Otherwise, we c* *an assume (after switching indices if necessary) that NG(K) = K3 or K23. If NG(K) * *= K23, then NG(K1; K12; K) K1 \ K23 = K. So suppose NG(K) = K3. Since K12 is not normal in G = , K12cannot be normal in both K1 and K2, and we can assume without loss of generality that K12is not normal in K1. Then NK1(K12) = K12, an* *d so N(K1; K12; K) K12\ K3 = K. This finishes the proof that (G; F) =2U2. It remainsftoishow that (1) always holds if K contains a Sylow p-subgroup for* * some prime pfi|G|. By definition of a regular basis, [Kij:K] > 1 for all i; j. If * *[Kij:K] = [Kik:K] = 2 for some i, then K C Ki= ([A2 , 28.1(2)]); Ki=K is genera* *ted by two elements of order 2 and hence dihedral; and this is a contradiction since i* *t means there are other overgroups of K not conjugate to any of the given ones. Thus, [Kij:K] = 2 for at most one pair of indices i; j. So if (1) does not ho* *ld, then the three indices [Kij:K] must be (2; 3; 3), (2; 3; 4), or (2; 3; 5). Since ea* *ch index is prime to p (K contains a Sylow p-subgroup), this shows that p 5. If [Kij:K] = * *m, then the permutation action of Kijon the set Kij=K restricts to a homomorphism 'ij:K ! m-1 4 whose kernel Rijis normal in Kij. Set H = O{2;3}(K) C K: the smallest normal subgroup of index a product 2r.3s. Then H is characteristic in* * any subgroup of K which contains H, and in particular characteristic in each Rij. S* *o H is normal in each Kij, and hence normal in G = . Since G is simple,* * H = 1, so 2 and 3 are the only primes dividing |K|. And thisfcontradictsithe assumptio* *n that_ K contains a Sylow p-subgroup for some p 5 and pfi|G|. * * |__| We are now ready to prove: Proposition 7.2. Let G be any of the sporadic simple groups, or the Tits group * *2F4(2)0. Then there is no 2-dimensional acyclic G-complex without fixed points. Proof.We first prove the proposition for ten of the sporadic groups as well as * *the Tits group, by direct application of Proposition 5.4. Since M22 is one of these* * groups, Proposition E.3 then applies to prove the proposition for the other four Mathie* *u groups. The last twelve sporadic groups are then handled using Lemma 7.1. Throughout t* *he proof, whenever two names are given for one of the sporadic groups, the first i* *s that used in [Atl], and the second the name used in [A1 ] or [A2 ]. Assume the proposition does not hold, and let G be the smallest such group wh* *ich has a fixed-point free action on a 2-dimensional acyclic complex X. Let F be the se* *parating family of subgroups H G such that XH 6= ;. Consider first the following table,* * which describes how Proposition 5.4 can be applied to these eleven simple groups: Bob Oliver and Yoav Segev * * 49 _______________________________________________________________________ || G | K1 | K2 | K3 | K12| K | K13 | K23 | [Kij:K] || ||____|______|________|_______|_____|________|_________|_______|_______ || || J1| 23:7:3| 7:6 | C3xD10| 7:3| 3 | 6 | 6 | 7;2;2 || || | | | | | | | | || || | | = N(7)| = C(3)| | | | | || ||____|______|________|_______|_____|________|_________|_______|______ || || M22| L3(4)| 24:S5| 24:A6| 24:A5| 22+4:3| 24:A5 | 22+4:S3|5;5;2 || || | | | | | | | | || || | (point)|(duad)| (hexad)| | = 24:A4| | = 24:S4| || ||____|______|________|_______|_____|________|_________|_______|______ || || J2| U3(3)|3.PGL2(9)|21+4:S3|31+2:8| 3:8 | 4.S4 | 3:D16| 9;4;2 || || | | | - | - | | | | || || | | = N(3A)| = N(4)| | | | | || ||____|______|________|_______|_____|________|_________|_______|______ || || HS| U3(5):2|U3(5):2|C(2x)| 51+2:8:2|5:8:2| 2S5:2 | 2S5:2| 25;6;6 || || | | | | + | | | | || || | | N(Syl5)| | | | | | || ||____|______|________|_______|_____|________|_________|_______|______ || || J3| L2(16):2|L2(17)| 21+4:S3|17:4| 4 | D8x2 | D16 | 17;4;4 || || | | | - | | | | | || || | | N(17)| = N(4)| | | | | || ||____|______|________|_______|_____|________|_________|_______|______ || || He| S4xL3(2)|L3(2)x7:3|N(3x)|S4x(7:3)|S4x3| S4xD6 | L3(2)x3| 7;2;7 || || | | | | | | | | || || | | = N(7)| | | | | | || ||____|______|________|_______|_____|________|_________|_______|______ || || Ru| 2F4(2)|L2(25).22|3.A6.22|L2(25).2|D24:2|(31+2:D8):2|D24:22|o;9;2 || || | | | | | | + | | || || | (point)|(edge)| = N(3)| | | | | || ||____|______|________|_______|_____|________|_________|_______|______ || || O'N| J1 | L3(7):2|(32xA6).2|19:6| 6 | D6xD10 | S3xS4| 19;10;24 || || | | | | | | | | || || | | N(19)| = N(3)| | | | | || ||____|______|________|_______|_____|________|_________|_______|______ || || HN| A12| M12:2| C(2x)| M12| 2xS5 | 25:S6 | (22xA5):2|o;96;2 || || | | | | | | | | || || = F5| (point)|(point|pair) | | = C (2A)|= C (2B)| | || ||____|______|________|_______|_____|____M12_|______A12|_______|______ || || Th| 3D4(2):3|(3x13):12|21+8:A9|13:12|12 | C (2x)| 3x12 | 13;o;3 || || | | | + | | | K1 | | || || = F3| | = N(13)| = C(2)| | | order 9216| | || ||____|______|________|_______|_____|________|_________|_______|______ || || 2F4(2)0|L2(25)|52:4A4|[29.3]| 52:12| 12 | D24 | 4A4 | 25;2;4 || || | | | | | | | | || || | | = N(Syl5)|C(2B)| | | | | || ||____|______|________|_______|_____|________|_________|_______|_______|| Table 5 We refer to [Atl] for the existence of subgroups with these properties, and to * *[GLS , Table 5.3] for tables of normalizers of prime order subgroups of the sporadic g* *roups. The subgroups in Table 5 are described using mostly the notation of [Atl]. How* *ever, we write, for example, N(3) or C(3) to denote the normalizer or centralizer in * *G of a subgroup of order 3 when there is a unique G-conjugacy class of such subgroups;* * and write N(3A) or N(3B) (or N(3x) when the class is unspecified) only when there i* *s more than one class. Also, Sylpalways denotes a Sylow p-subgroup of G. In all cases, the results of Section 6 and Appendix F and the minimality assu* *mption on G imply that Ki 2 F for all i = 1; 2; 3. Note in particular the cases G = HS* * , He, and HN: K3 2 F since K13or K23is nonsolvable and in F. The remarks under the names of the subgroups Ki describe how they are chosen relative to one another. In all cases except G = M22, K1 and K2 are chosen in o* *ne of the following two ways: either (a) they are the stabilizers of a vertex and an edge (or point pair) of a stand* *ard action of G on a graph; or (b) K1 is a maximal subgroup of G, and K2 is the normalizer of some subgroup X * * K1 (as indicated in the table), or a maximal subgroup (not conjugate to K1) contai* *ning NG(X) and such that K12= NK1(X). 50 Fixed point free actions on acyclic 2-complexes The subgroup K3 is then chosen as the normalizer or centralizer of a certain su* *bgroup Y K12 as indicated. In all cases where K12 contains more than one conjugacy cl* *ass of subgroup of the given order, the choice is either specified under K = NK12(Y* * ), or is clear from the description of K. In many cases, it is unnecessary to identify K* *3 more precisely, since the only thing we need know about it is that it must lie in F. When G = M22, K3 ~=24:A6 is the subgroup which leaves invariant some hexad in the Steiner system of order 22, and it has the obvious action on this set of or* *der 6 (cf. [Gr , Theorem 6.8]). Then K1 is taken to be the stabilizer of some point x in t* *he hexad, and K2 the stabilizer of some pair of points in the hexad including x. In all cases, each of the subgroups in the sequence K K12 K1 G is maximal and self-normalizing in the next one. Thus, conditions (b,c,d) in Proposition 5* *.4 always hold. Condition (e) ((K1; K12; K) is not G-conjugate to (K1; K13; K)) is clear* * except when G = M22; in this case K12and K13are distinct parabolic subgroups in K1 ~=L* *3(4) containing the same Borel subgroup K, and hence not conjugate in K1. Inequality* * (a1) holds in all cases except when G = J1, as can be checked using the list of indi* *ces [Kij:K] in the last column (where "o" means that the index is >10 and hence large enoug* *h not to matter). We give particular attention to the case G = J1: the first Janko group, and * *the only sporadic group not handled in [AS ]. Fix some K1 ~=C32o(C7o C3): a maxim* *al subgroup of G by [A2 , 16.17] (see also 16.4 and 16.16 in [A2 ]). Let K2 ~=C7o* * C6 be the normalizer of a subgroup of order 7 in K1, and let K3 ~=C3x D10be the centr* *alizer in G of a subgroup of order 3 in K12. Then K12 ~=C7o C3, K13 ~=C6 ~= K23, and K = K1 \ K2 \ K3 ~=C3. All of these subgroups are solvable, and hence in F. Als* *o, X 1 1 1 1 1 1 1 1 _______= __+ __+ __< 1 + ___+ ___= 1 + _______+ _______; i 0, 'm (Jm xSm-1 ) X(m-1). Moreover, for each j 2 Jm , there* * are finite subsets J0k Jk (0 k m - 1) such that m-1[ 'm (jx Sm-1 ) 'k(J0kxDk): k=0 (c) A subset U X is open if and only if '-1m(U) is open in Jm xDm for each m* * 0. (X has the "weak topology" with respect to its cell structure.) In the above definition, G is always assumed to act trivially on Dm and Sm-1 * *. Usually, a G-CW complex will be called a G-complex for short. A CW complex is just a G-CW complex in the case where G is the trivial group. An open cell in a (G-)CW complex X is the image 'm (jx int(Dm )) of one open di* *sk under the characteristic_map. Note that if oe = 'm (jx int(Dm ))_is any open ce* *ll, then 'm (jx Dm ) = oe(the closure of oe) and @oe = 'm (jx Sm-1 ) = oer oe (the bound* *ary of oe) are determined by oe itself as a subspace of X. By condition (a) in the definit* *ion, each point of X lies in exactly one open cell, and the open m-cells of X are the con* *nected components of X(m)r X(m-1). The following is an alternative way to regard G-complexes, once CW complexes * *have been defined. Fix a CW complex X with continuous G-action. Call the action admi* *ssible if it permutes the open cells of X, and sends a cell to itself only via the ide* *ntity. If X is a G-complex, then by definition the G-action is admissible. Conversely, if t* *he action of G on X is admissible, then the characteristic maps of X can be redefined to * *yield a G-complex. More precisely, if 'm : Jm xDm ! X is the given characteristic map f* *or the m-cells of X, then the action of G on the m-cells of X induces an action on Jm * *. Also, for any orbit of G on Jm and any j 2 , one can define '0mon x Dm by setting '0m(gj; x) = g'm (j; x). Upon doing this for all m 0 and all orbits of Jm , we* * get the new characteristic maps which make X into a G-complex. Note in particular the last part of condition (b). Each cell in a CW complex* * must be "closure finite": its boundary must be contained in a finite union of closed* * cells of smaller dimensions. To see the importance of this condition, consider the space* * X = D2, let J0 be the circle S1 with the discrete topology, let J2 be a set with one el* *ement, and set Jm = ; for all m 6= 0; 2. Let '0: J0x D0 ! X be projection to the first fac* *tor (i.e., inclusion of the circle), and let '2: J2x D2 ! X be projection to the second fa* *ctor. These sets and maps satisfy all of the conditions for a CW structure on D2 exce* *pt for closure finiteness. But this goes against our intuitive expectations (by analo* *gy with simplicial complexes) that the 0-skeleton of any CW complex should be discrete,* * and that compact CW complexes should be made up of finitely many cells. The following lemma describes the principal means of constructing G-complexes* * (see, e.g., Proposition 0.2). Lemma A.2. Let X be a G-complex, let J be any discrete set with G-action, and* * let ': Jx Sn-1 ! X(n-1)be any G-equivariant map. Then the space Y = X [' (Jx Dn) Bob Oliver and Yoav Segev * * 53 is a G-complex. Proof.Let : Jx Dn ! Y be the obvious map; thus |Jx Sn-1= '. This, together with the characteristic maps for X, make up the characteristic maps for Y . The othe* *r details_ are the same as in the nonequivariant case; cf. [LW , Proposition II.2.2]. * * |__| If X has been constructed via successively attaching cells, i.e., via success* *ive repetition of the construction of Lemma A.2, starting with a discrete set, then the closur* *e finiteness condition holds automatically. In fact, this is the basis for an alternative de* *finition of a CW complex, described more precisely in [LW , Theorem II.2.4]. A (G-invariant) subcomplex of a G-CW complex is a closed (G-invariant) subspa* *ce A X which is a union of closed cells in X; i.e., a union of images of characte* *ristic maps. A subcomplex is itself a CW complex in an obvious way. Note in particular* * that if X is a G-complex, then for every H G, the fixed point set XH is a subcomple* *x of X: if 'm : Jm xDm ! X are the characteristic maps for X, then ('m )H : (Jm )H xDm * *! XH are the characteristic maps for XH . The following proposition is an immediate consequence of condition (c) in Def* *inition A.1. Roughly, it says that a function defined on a CW complex is continuous if* * and only if its restriction to each closed cell of the complex is continuous. Lemma A.3. Let X be a CW complex, with characteristic maps 'm : Jm xDm ! X. Then if Y is any topological space, a function f : X ! Y is continuous if and o* *nly_if f O'm is continuous for each m. |* *__| Recall (cf. [S1], [AS ]) that a simplicial complex X with G action is called * *admissible if the action permutes the simplices linearly, and sends a simplex to itself on* *ly via the identity. (If this last condition does not hold, then it does hold for the bar* *ycentric subdivision of X.) We claimed in the introduction that Theorem A holds equally * *well if one replaces "G-complex" by "admissible G-simplicial complex" in the statement.* * This follows from the following proposition, where simplicial complexes are always a* *ssumed to have the metric topology (cf. [LW , Definition IV.4.1]). Proposition A.4. Any finite dimensional admissible G-simplicial complex is G-ho* *mo- topy equivalent to a G-complex of the same dimension. Any countable, finite di* *men- sional G-complex is G-homotopy equivalent to an admissible G-simplicial complex* * of the same dimension. Proof.For any admissible G-simplicial complex X, one can clearly define skeleta* * and characteristic maps for X which satisfy conditions (a) and (b) in Definition A.* *1; but for these to also satisfy condition (c) we must replace X with a new space Xcw havi* *ng the same underlying set but a finer topology (more open sets). The identity map Xcw* * ! X is continuous and is a homotopy equivalence by [LW , Proposition IV.4.6] (and* * the argument in [LW ] can easily be fixed to cover the equivariant case). The second statement is shown, in the nonequivariant case, by Whitehead in [W* *h , Theorem 13], and his proof carries over immediately to G-complexes. The idea is* * the following: once X(m-1)has been replaced by a G-simplicial complex of the same d* *imen- sion, then approximate the characteristic map Jm xSm-1 ! X(m-1)by a simplicial * *map (possibly after further subdivision of X(m-1)), and attach the m-cells after gi* *ving_them appropriate simplicial structure. * * |__| 54 Fixed point free actions on acyclic 2-complexes For any space X, we let X denote its unreduced suspension: X def=(Xx I)=~ , where (x; 0)~ (x0; 0) and (x; 1)~ (x0; 1) for all x; x02 X. A G-action on X aut* *omatically determines a G-action on X, via the trivial action on the interval I. Lemma A.5. The orbit space X=G of a G-complex X inherits a structure of a CW complex, with one n-cell in X=G for each G-orbit of n-cells in X. The unreduced suspension X of any G-complex X is itself a G-complex in a natural way. And if X and Y are any two G-complexes, at least one of which is finite, then their prod* *uct Xx Y is also a G-complex. Proof.If X is a G-complex, with skeleta X(m) and characteristic maps 'm : Jm xS* *m ! X, then X=G is a CW complex with skeleta (X=G)(m)= X(m)=G and characteristic ma* *ps 'm =G: (Jm =G)x Sm ! X=G. This follows immediately from Definition A.1. Note* * in particular that condition (c) holds for X=G by definition of the quotient topol* *ogy: a subspace is open in X=G if and only if its inverse image is open in X. The unreduced suspension of a CW complex is again a CW complex by [LW , Corol* *lary II.5.12]. And if X or Y is finite, then Xx Y is a G-complex with the obvious pr* *oduct structure by [LW , Theorem II.5.2]. In each of these last two cases, the argum* *ents in [LW ] carry over without change to the equivariant case. We remark here that if X and Y are arbitrary CW complexes,Sthen there is an obvious way to define skeleta for Xx Y : (Xx Y )(m) = i+j=m(X(i)xY (j)). Al* *so, if 'm : Jm xDm ! X and m : Km xDm xY are the`characteristic maps for X and Y , th* *en one can define characteristic maps !m = i+j=m('ix j) for Xx Y . (This require* *s fixing identifications DixDj ~=Di+j.) Conditions (a) and (b) in Definition A.1 always * *hold;_ what can go wrong is condition (c). * *|__| The following lemma is not used in the paper, but does help to motivate the c* *oncept of "universal" (G; F)-complexes as defined in Section 0. Proposition A.6. Fix a family F of subgroups of G, and let Y be any universal (* *G; F)- complex. Then for any (G; F)-complex X, any G-invariant subcomplex A X, and any equivariant map f0: A ! Y , f0 extends to an equivariant map f : X ! Y . Furthe* *rmore, f is unique up to homotopy, in the sense that if f0: X ! Y is any other extensi* *on of f0, then there is an equivariant homotopy F : Xx I ! Y such that F |Xx0 = f, F |Xx1* * = f0, and F |AxI = f0OprojA. Proof.It suffices to prove the existence of f : X ! Y ; the uniqueness then fol* *lows by extending the given map on (Xx {0; 1}) [ (Ax I) to Xx I. We construct f : X ! Y one skeleton at a time. The construction of f0: X(0)[A* * ! Y is easy: let {xi} be orbit representatives for the vertices not in A, set Hi = * *Gxi (the isotropy subgroup), choose any yi2 Y Hi, and define f0(gxi) = gyi for all g 2 G* * and all i (and f0|A = fA). Now assume that n 1, and that fn-1: X(n-1)[ A ! Y has been constructed. Let 'n: Jnx Dn ! X be the characteristic map for the n-cells of X (where Jn is a di* *screte set with G-action), and let J0n Jn be the subset of those n-cells not in A. Set u0 = fn-1O'n|J0nxSn-1: J0nxSn-1 -----! Y: Bob Oliver and Yoav Segev * * 55 For each j 2 J0n, let Gj = {g 2 G | gj = j} 2 F be its isotropy subgroup. Then u0(jx Sn-1) Y Gj. Also, Y Gjis contractible (since Y is (G; F)-universal), the* * identity map Y Gj! Y Gjis homotopic to a constant map, and hence any map to Y Gjis homo- topic to a constant map. In particular, u0 can be extended to a (nonequivariant* *) map v0j: jx Dn ! Y Gj. This can then be extended to a G-map vj: Gjx Dn ! Y (where G* *j is the orbit of j) by setting vj(gj; x) = g.v0j(j; x). Upon repeating this procedu* *re with one representative from each G-orbit in J0n, the vj combine to give a G-map u: J0nx* *Dn ! Y whose restriction to J0nxSn-1 is u0. If we now set fn(x) = fn-1(x) for x 2 X(n-* *1)[ A, and fn('n(j; x)) = u(j; x) for (j; x) 2 J0nxDn, then this is a well defined map* * of sets_from X(n)to Y , which is equivariant by construction, and continuous by Lemma A.3. * * |__| Note that Proposition A.6 implies in particular that any two universal (G; F)* *-com- plexes are G-homotopy equivalent. Appendix B. Cellular homology of G-complexes The cellular chain complex (Cn(X); @n)n0 of a CW complex is described in [LW* * , xV.2]. Formally, this is defined using singular homology (in particular, Cn(X)* * = Hn(X(n); X(n-1))), as in [LW , Definition V.2.1]. By [LW , Proposition V.1.8],* * Cn(X) is the free abelian group with basis the set of (oriented) n-cells in X; and by* * [LW , xV.3] each boundary map @n: Cn(X) ! Cn-1(X) can be described via the matrix whose en- tries are the degrees of maps between (n - 1)-spheres induced by the attaching * *maps for the n-cells. By [LW , Theorem V.2.11], the singular homology H*(X) is isomo* *rphic to the homology of the complex (Cn(X); @n). Hence, if X is a finite complex, th* *e Eu- ler characteristic O(X) is equal to the alternating sum of the numbers of cells* * in each dimension. Note that for a map f : X ! Y between CW complexes to induce a homomorphism C*(X) ! C*(Y ), it must be a cellular map, in the sense that f(X(n)) Y (n)for * *all n 0. However, since cellular homology H*(C*(X); @) is isomorphic to singular homo* *logy, any continuous map between CW complexes induces a homomorphism between their cellular homology groups. More generally, if X is any CW complex and A X is any subcomplex, then the relative cellular chain complex is defined by setting C*(X; A) def=C*(X)=C*(A).* * Thus, Cn(X; A) is the free abelian group with one generator for each n-cell of X not * *in A. By [LW , Theorem V.2.11] again, the homology of the complex (C*(X; A); @) is natur* *ally isomorphic to H*(X; A). If X is a G-complex and A X is a G-invariant subcomplex, then the cellular c* *hain complexes C*(X) and C*(X; A), and the homology groups H*(X) and H*(X; A), are a* *ll Z[G]-modules. In fact, each chain group Ci(X) or Ci(X; A) is a permutation modu* *le, in the sense that it has a Z-basis which is permuted by the linear action of G. Once homology has been defined using the cellular chain complex, then the rel* *ative and Mayer-Vietoris exact sequences, and excision, are immediate. (Note, however* *, that excision in singular homology is needed to establish the basic properties of ce* *llular homology of CW complexes [LW , xV.1-2].) To see this, fix a G-complex X. For * *any 56 Fixed point free actions on acyclic 2-complexes G-invariant subcomplexes A0 A X, the short exact sequence of chain complexes 0 ---! C*(A)=C*(A0) ----! C*(X)=C*(A0) ----! C*(X)=C*(A) ---! 0 induces, via the snake lemma, the relative exact sequence . .-.--!Hi(A; A0) ----! Hi(X; A0) ----! Hi(X; A) --@--!Hi-1(A; A0) ---! . .:. Similarly, for any pair of G-invariant subcomplexes A; B X with A [ B = X, the* *re is a short exact sequence 0 ---! C*(A \ B) ----! C*(A) C*(B) ----! C*(A [ B) ---! 0 which induces the Mayer-Vietoris sequence . . .---!Hi(A \ B) ---! Hi(A) Hi(B) ---! Hi(X) --@-!Hi-1(A \ B) ---! . .:. All of these are exact sequences of Z[G]-modules. Similarly, since Br (A\B) contains exactly the same cells as (A[B)r A, the in* *clusion map induces an isomorphism ~= H*(B; A \ B) -----! H*(A [ B; A) (excision) since it induces an isomorphism of cellular chain complexes. The following lemma, used in the proof of Proposition 0.2, is one application* * of excision and the relative exact sequence. It describes the effect of attaching * *cells on the homology of the complexes involved. Lemma B.1. Let X be a G-complex, let J be a discrete set with G-action, and l* *et f : Jx Sn ! X(n)be any G-equivariant map (n 1). Set Y = X [f (Jx Dn+1). Then there is an exact sequence of Z[G]-modules 0 ---! Hn+1(X) -incl*---!Hn+1(Y ) ----! f* incl* Hn(Jx Sn) ----! Hn(X) ----! Hn(Y ) ----! 0; and the inclusion X -incl-!Y induces isomorphisms Hi(X) ~=Hi(Y ) for all i 6= n* *; n + 1. Proof.Let ff: Jx Dn+1 ! Y be the characteristic map (so ff|Jx Sn= f). This indu* *ces an isomorphism C*(Jx Dn+1; Jx Sn) ~=C*(Y; X) of chain complexes, and hence an i* *so- morphism in homology in all degrees. The following square Hn+1(Jx Dn+1; Jx Sn) --@-!~Hn(Jx Sn) ? = ? ff*?y~= Hn(f)?y Hn+1(Y; X) --@-! Hn(X) commutes by the naturality of the relative exact sequences for pairs of CW comp* *lexes, and the upper boundary map is an isomorphism since Hi(Jx Dn+1) = 0 for i 1. The lemma now follows from the relative exact sequence for the pair (Y; X), whe* *re__ Hn+1(Y; X) is replaced by Hn(Jx Sn) via the above square. * * |__| The following more technical application of excision and the relative exact s* *equences is needed in the proof of Proposition 1.7. Bob Oliver and Yoav Segev * * 57 Lemma B.2. Fix a CW complex Y and subcomplexes B; X Y , and set A = B \ X. Assume that the inclusion induces an isomorphism H*(A) ! H*(B). Then H*(Y; X) ~= H*(Y; X [ B). Proof.It suffices to show that H*(X [ B; X) = 0; the result then follows from t* *he relative exact sequence for Y X[ B X. But H*(X [ B; X) ~=H*(B; A) by excision* *, __ and this last group vanishes since H*(A) ~=H*(B). * * |__| The following result says, roughly, that a union of homology or homotopy equi* *valences between CW complexes is again a homology or homotopy equivalence. Proposition B.3. Let f : X ! Y be a map between CW complexes. Fix subcomplexes A1; A2 $ X and B1; B2 Y such that X = A1 [ A2 and Y = B1 [ B2, and set A0 = A1 \ A2 and B0 = B1 \ B2. Assume that f restricts to homology (homotopy) equivalences fi: Ai ! Bi for i = 0; 1; 2. Then f is itself a homology (homotop* *y) equivalence. Proof.If f0, f1, and f2 are all homology equivalences, then f is a homology equ* *ivalence by the Mayer-Vietoris sequences for the two unions (and the 5-lemma). Assume now that f0, f1, and f2 are all homotopy equivalences; we must show th* *at f is a homotopy equivalence. By the Van Kampen theorem, f induces an isomorphism * *of fundamental groups (on each connected component). The map between the universal covers is a homology equivalence, hence a homotopy equivalence; and hence f : X* * ! Y is itself a homotopy equivalence. For the details of this argument, cf. [Gra * *, Lemma 16.24 & Theorem 16.22]. Alternatively, and more geometrically, one can show directly that any homotop* *y in- verse g0: B0 ! A0 of f0 can be extended (one cell at a time) to homotopy invers* *es gi: Bi! Ai(i = 1; 2), while at the same time extending the homotopies of g0Of0 * *' IdA0 and f0Og0 ' IdB0. The result then follows upon taking g = g1 [ g2: Y ! X (and s* *imi- larly for the homotopies). The existence of the gi and the homotopies follows f* *rom the proofs of [LW , Theorems IV.3.2-3] (applied to the 2-ads (Ai; A0) and (Bi; B0))* *; although_ the statements of these theorems are not sufficiently precise to do this. * * |__| Appendix C. Projective Z[G]-modules Recall that for any G-complex X, C*(X) and H*(X) are Z[G]-modules in an obvio* *us way. A finitely generated Z[G]-module M will be called stably free if there are* * finitely generated free modules F0 and F such that M F0 ~=F . Free Z[G]-modules, and he* *nce (as an intermediate step) stably free Z[G]-modules play a key role when constru* *cting finite G-complexes in Section 1. Lemma C.1. If X Y are finite G-complexes such that G acts freely on Y rX, th* *en C*(Y; X) is a finite chain complex of free finitely generated Z[G]-modules. Proof.By assumption, G permutes freely the cells in Y not in X. Thus, G permutes freely a basis of C*(Y; X); and this is a finite basis since X and Y have only* *_finitely many cells. |_* *_| 58 Fixed point free actions on acyclic 2-complexes The following lemma says in particular that if C* is a finite chain complex o* *f finitely generated free Z[G]-modules all but one of whose homology groups is stably free* *, then the remaining homology group is also stably free. This does not hold for module* *s over arbitrary noetherian rings, but uses special properties of group rings. Proposition C.2. Let C* be any finite chain complex of projective Z[G]-modules.* * As- sume, for some k, that Hi(C*) is projective as a Z[G]-module for all i 6= k, an* *d that Hk(C*) is Z-free. Then Hk(C*) is also a projective Z[G]-module, and M M M M Hi(C*) Ci~= Hi(C*) Ci: (1) i even i odd i odd i even Proof.We first claim the following: if 0 ! A ! B ! C ! 0 is a short exact seque* *nce of finitely generated Z-free Z[G]-modules, and two of the modules A, B, and C a* *re projective (stably free), then so is the third. This is clear if C is projectiv* *e, since in that case B ~=A C. So assume that A and B are projective (stably free). Since all t* *hree groups are Z-free and finitely generated, the dual sequence 0 ! C* ! B* ! A* ! * *0 is also exact. Here, for any Z[G]-module M, M* def=HomZ(M; Z) has the obvious stru* *cture as a Z[G]-module. Dualization clearly takes finitely generated free Z[G]-module* *s to free Z[G]-modules, hence the same for projective modules; and so the dualized sequen* *ce splits. Thus B* ~= A* C* as Z[G]-modules; and upon dualizing again we see that B ~=A C. So C is Z[G]-projective (stably free). Now fix any m; n 2 Z such that m < k < n, and Ci= 0 for all i < m and all i >* * n. For each i, set Zi = Ker[Ci--@! Ci-1] and Bi = Im[Ci+1--@! Ci]. Consider the sh* *ort exact sequences 0 --! Zi---! Ci---! Bi-1--! 0 and 0 --! Bi---! Zi---! Hi(C*) --! 0: By induction starting at i = m, one sees that Zi is projective for each i k, a* *nd that Biis projective for each i < k. Similarly, by downward induction starting at i * *= n + 1, one sees that Biis projective for each i k, and that Ziis projective for each * *i > k. In particular, Bk and Zk are both projective, and so the same holds for Hk(C*). In particular, theLabove shortLexact sequencesLsplit, since all ofLtheir term* *s are pro- jective. Set Cev= (C2i), Cod= (C2i+1), Hev= (H2i(C*)), Hod= (H2i+1(C*)); and similarly for Zev, Zod, Bev, and Bod. Then Hev Cod~= Hev Bev Zod~= Zev Zod; Hod Cev~=Hod Bod Zev~=Zod Zev; * *__ and this proves (1). |* *__| The following property of projective Z[G]-modules is a consequence of a theor* *em of Swan. Proposition C.3. Let P and P 0be any two finitely generated projective Z[G]-mod* *ules. Then P Z P 0is stably free as a Z[G]-module. Proof.Assume first that P is free. Let {ai} be a Z[G]-basis for P , and let {b* *j} be a Z-basis for P 0. Then {ai bj} is a Z[G]-basis for P P 0, and this module is fr* *ee. (Note that we did not need to know that P 0is projective, only that it is Z-free.) Bob Oliver and Yoav Segev * * 59 Now consider the general case. By [Sw , Theorems 7.1 and 8.1], for any n > 0,* * any finitely generated projective Z[G]-module contains a free submodule of finite i* *ndex prime to n. In particular, we can choose free submodules F P and F 0 P 0, such that * *[P :F ] and [P 0:F 0] are finite and relatively prime. Consider the commutative diagram 0 ---! F F 0 -i1--!P F 0 ---! (P=F ) F 0---! 0 ? ? ? j1?y j2?y ff?y~= 0 ---! F P 0 -i2--!P P 0 ---! (P=F ) P 0---! 0; where all tensor products are taken over Z. The rows are both exact, and ff is* * an isomorphism since (P=F ) (P 0=F 0) = 0. So by an easy diagram chase, the seque* *nce (i1;j1) 0 0 j2-i2 0 0 ---! F F 0------! (P F ) (F P ) ------! P P ---! 0 is exact. We have just seen that the first two terms in this sequence are free* *,_and so P P 0is stably free. |* *__| In fact, using stability results of Swan, one can show that the tensor produc* *t of any two finitely generated projective Z[G]-modules is free. This is not needed * *for the constructions in this paper, but the following much deeper stability result is * *used. It is not needed to prove the existence of 2-dimensional acyclic G-complexes, but it * *is used in Section 3 to show that all of the complexes we construct can be taken to hav* *e exactly one free orbit of 2-cells (and no free orbits of cells in other dimensions). Proposition C.4. If G is simple, or (more generally) if there is no homomorphism G ! SU(2) (= SU(2; C)) with nonabelian image, then any stably free Z[G]-module * *is free. Proof.By a theorem of Jacobinski [Jac, Theorem 4.1], if A is any Z-order in a f* *inite di- mensional semisimple Q-algebra A which satisfies the Eichler condition, then al* *l finitely generated stably free A-modules are free. Here, the algebra A satisfies the Eic* *hler con- dition if it has no simple factor B, with center K, for which every embedding K* * ,! C has image contained in R and induces an isomorphism R K B ~= H (the quaternion algebra over R). If Q[G] does not satisfy the Eichler condition _ if B is a simple summand of * *Q[G] and R K A ~=H _ then the composite proj Q[G] -----! B -----! H restricts to a multiplicative homomorphism ff: G ! S3 ~=SU(2; C). Here, S3 deno* *tes the group of quaternions of norm 1. And since the image of G in H generates H a* *s an __ R-vector space, Im(ff) must be nonabelian. See also [Re , x38] for more discuss* *ion. |__| Appendix D. Finite simple groups of Lie type We give here a very short discussion of groups of Lie type. For more detail, * *we refer to [St1], [St2], [Ca ], or [GLS ]. 60 Fixed point free actions on acyclic 2-complexes The finite simple groups of Lie type consist of the Chevalley groups and thei* *r twisted analogs. The finite Chevalley groups are analogs of the (complex or compact) Li* *e groups, but realized over a finite field. They thus include the four families of classi* *cal groups: An(q) ~=Ln+1(q) = P SLn+1(q), Bn(q) ~=P 2n+1(q) (the commutator subgroup of the projective orthogonal group P GO2n+1(q)), Cn(q) ~= P Spn(q), and Dn(q) ~= P +2n* *(q) (the commutator subgroup of the projective special orthogonal groups with respe* *ct to a quadratic form of "plus type"); as well as the exceptional groups E6(q), E7(q* *), E8(q), F4(q) and G2(q). All of these are defined over any finite field; i.e., for any * *prime power q. The finite twisted groups of Lie type were first treated systematically by St* *einberg in [St1] and [St2], where (very roughly) they are obtained a s fixed points of * *certain automorphisms of the Chevalley groups _ group automorphisms which are associated with automorphisms of the Dynkin diagram. Let G be one of the symbols An; Bn; C* *n, etc. Then mG(q) denotes the fixed subgroup of an automorphism of order m of G(q* *m ) (or of G(q) when G = B2, G2, or F4). The finite twisted groups thus consist of* * the classical groups 2An(q) ~=P SUn+1(q) = Un+1(q) and 2Dn(q) ~=-2n(q) (the commuta* *tor subgroup of the projective special orthogonal groups of "minus type"); as well * *as the Suzuki groups 2B2(22k+1), the Ree groups 2G2(32k+1) and 2F4(22k+1), and the Ste* *inberg groups 2E6(q) and 3D4(q). To make this more concrete, it is necessary_to work with automorphisms_of the* * Cheval- ley groups over the algebraic closure Fp,_where p is prime. Let G(F p) denote_a* * simple algebraic group of type G defined over Fp. We will always assume that G(F p) is* * of ad- joint type (i.e., with trivial center), or equivalently that it is a group of a* *utomorphisms of the corresponding Lie algebra. For q a power of p, the finite Chevalley grou* *p_G(q) can (roughly) be thought of as the fixed subgroup of the automorphism 'q of G(F* * p) induced_by_the field automorphism (t 7! tq). More generally,_a_Steinberg endomo* *rphism of G = G(F p) is_defined to be an algebraic endomorphism of G whose fixed subgr* *oup C_G(oe) = {x 2 G | oe(x) = x} is finite. (In fact, the Steinberg endomorphisms * *are all __ automorphisms of G as an abstract group, but none of them is invertible as an a* *lgebraic endomorphism.) The finite twisted_groups of Lie type are (roughly) the fixed su* *bgroups of Steinberg endomorphisms of G, which are field automorphisms (t 7! tq) "twist* *ed" by graph automorphisms. __ __ __ More precisely, if oe is a Steinberg endomorphism of G = G(F p), let_Goedenot* *e the sub- group of C_G(oe) generated by its Sylow p-subgroups. Equivalently, Goe= , __ where U; V G are subgroups_defined_in the next paragraph. If eGis the universa* *l cen- tral extension of G, then Goe~=CGe(oe)=Z, where Z denotes the center. For examp* *le, if q is a power of p, then SLn(q) = CSLn(_Fp)('q), while P SLn(q) can be a proper su* *bgroup __ of CPSLn(_Fp)('q). For all G and all q = pk, G(q) = G'q. To describe the Steinberg endomorphisms, we must first establish_notation for* * certain elements of the Chevalley groups. Fix a prime p, and let F Fp be any subfield.* * Set G = G(F), and let be the system of roots of type G. Let +; - denote the sets of positive and negative roots, respectively. To each r 2 there corresponds a * *subgroup (the root subgroup) Xr = {xr(t) | t 2 F} G, isomorphic to the additive group F* *. Then Bob Oliver and Yoav Segev * * 61 U def= and V def= are_both maximal unipotent subgroups of G; they are closed and connected if F = Fp, and are Sylow p-subgroups of G if F* * is finite._Also,_G = . The subgroup H def=NG(U)\ NG(V ) is a maximal torus * *of G if F = Fp, and is called a Cartan subgroup of G when G is finite. This subgrou* *p H is abelian, generated by elements hr(t) for simple roots r and t 2 F*; and its * *elements are called "diagonal elements" of G. Also, NG(U) = UH and NG(V ) = V H (the Bor* *el subgroups of G). For example, when G = An(F) ~=Ln+1(F), (of adjoint type), then the roots corr* *espond to the pairs (i; j) for i 6= j, and the positive roots correspond to the pairs * *(i; j) for i < j. In this case, xij(t) = eij(t), the matrix which has 1's on the diagonal, t in p* *osition (i; j), and zeros elsewhere. Thus U and V are the subgroups of (strict) upper a* *nd lower triangular matrices, and H is the subgroup of diagonal matrices. Note that whe* *n we describe elements and subgroups here in terms of matrices, we mean their images* * under the surjection of SLn+1(F) onto Ln+1(F) = P SLn+1(F). __ __ Let oe be a Steinberg endomorphism of G = G(F p) (still assumed_of adjoint ty* *pe)._By the Lang-Steinberg theorem [St2, Theorem 10.1],_for any g 2 G, there exists_h_2* * G such that g = oe(h)h-1. Hence, all elements in Inn(G ) Ooe are conjugate in Aut(G_).* * In other words, composing_a Steinberg endomorphism oe with an inner automorphism of G, d* *oes not change G oe(up to conjugation). __ Next, Steinberg showed that for any oe, there is some g 2 G such that conj(g* *) Ooe leaves U and V invariant, and permutes the root subgroups Xr. It thus suffice* *s to consider those oe for which oe(Xr) = Xae(r)for some automorphism ae of the root* * system of type G, which preserves the positive roots; i.e., a permutation of which p* *reserves angles between the roots, such that ae(+) = +. Hence ae permutes the simple roo* *ts, and induces a symmetry of the Dynkin diagram of G. By inspection of the Dynkin diag* *rams, one sees that if ae 6= Id, then either G = An, Dn, or E6 and ae is the automorp* *hism of order 2 of the root system; or G = D4 and ae is an automorphism of order 3; or * *G = B2, F4, or G2 and ae is an automorphism of order 2 which interchanges long and shor* *t roots. * * __ If oe(Xr) = Xae(r)for such ae, then necessarily oe(xr(t)) = xae(r)(fflrtqr) f* *or some fflr 2 (F p)* and some qr powers of p. After composing with conjugation by a diagonal element* *, we can assume fflr = 1 for all simple roots r (and fflr = 1 for all r). Also, by s* *tudying the action of oe on diagonal elements, one can show that the ratio qr.krk=kae(r)k i* *s constant, independent of r. In particular, if ae = Id, then oe = 'q (q = qr for all r) i* *s a field automorphism. Assume that ae 6= Id, and that all roots in have the same length. Then oe = * *'qO ae, where q = pk >_1_(q = qr for all r); and where ae(xr(t)) = xae(r)(t) for all s* *imple roots r and all t 2 Fp (and ae(xr(t)) = xae(r)(t) for arbitrary r). The existence of* * such an automorphism aeis shown in [St1, Theorem 29] or [Ca , Proposition 12.2.3]. If * *m is the __ def__ order of ae, then oem = 'qm, so G oem= G(qm ), and mG(q) = Goecan be viewed as* * the subgroup of CG(qm()o) generated by its Sylow-p subgroups, where o is the restri* *ction of oe to G(qm ). In other words, we can regard mG(q) = G(qm )o, where o is the* * field automorphism (t 7! tq) "twisted" by the "graph automorphism" of G(qm ). 62 Fixed point free actions on acyclic 2-complexes As one example, consider the automorphism o(aij) = (((-1)i+jan+2-j;n+2-i)q)-1* * of Ln+1(q2). This preserves upper and lower triangular matrices, and sends xij(t)* * to xn+2-j;n+2-i(tq). The signs have been chosen so that o(xr(t)) = xae(r)(tq) whe* *n r is a simple root (i; i + 1) (but not for all roots). Then 2An(q) def=(Ln+1(q2))o =* * P SUn+1(q) is the projectivePspecial unitary group defined with respect to the hermitian f* *orm (x; y) = u. (-1)i+1xi(yn+2-i)q on (Fq2)n+1 (where u = 1 if n is even and uq-1* *= -1 if n is odd). Note that there can be elements of P SLn+1(q2) fixed by o which are * *not rep- resented by unitary matrices, which is why one must define 2An(q) = . If one works in the universal central extension SLn+1(q2) , the subgroup of elemen* *ts fixed by o is SUn+1(q). If has roots of distinct lengths and ae is nontrivial, then as mentioned abo* *ve G = B2, F4, or G2 and ae interchanges long and short roots. Set p0 = 2 if G = B2; F4 an* *d p0 = 3 if G = G2, so that kae(r)k_krk= (p0)1=2 for each r 2 . Since qr._krk_kae(r)kis * *independent of r (and the qr all powers of p), this is possible only if p = p0. Hence, oe = 'qO * *aefor some q = pk 1, where ( xae(r)(tp)if r is a short root ae(xr(t)) = xae(r)(t)if r is a long root. __ def__ Then oe2 = 'q2p, so G oe2= G(q2p) = G(p2k+1), and 2G(p2k+1) = Goecan be regard* *ed as the fixed subgroup of an involution on G(p2k+1). This group is sometimes den* *oted 2G(pk+1_2). As an example, Ono [On ] carried out this procedure on Sp4(22k+1) = B2(22k+1), regarded as the group of 4x 4 matrices which preserve the symplectic form (x; y* *) = x1y4 + x2y3 + x3y2 + x4y1. He obtained precisely the matrix presentation of Sz(* *22k+1) described in Proposition 3.6, as the fixed points (Sp4(22k+1))o, where o is the* * restriction of the above oe = 'qO aeto Sp4(22k+1). __ __ The rank of a Chevalley group G(q) is just the rank of G = G(F p) in the usu* *al sense; i.e., the number of simple roots in its root system, or the number of no* *des in its Dynkin diagram. The rank of a twisted group mG(q) is equal to the number of orb* *its of roots (or of nodes) under the corresponding automorphism of the root system * *or the Dynkin diagram of G. There are thus four families of finite simple groups of Li* *e type and Lie rank 1: the two dimensional projective special linear groups L2(q) ~=A1* *(q), the three dimensional projective special unitary groups U3(q) ~=2A2(q), the Suzuki * *groups Sz(q) ~=2B2(22k+1), and the Ree groups Ree(32k+1) ~=2G2(32k+1). We now return to the internal structure of the groups of Lie type. First let * *G = G(F ) be a Chevalley group over any field F , and let be a root system of type G. We* * have already discussed the root subgroups Xr = {xr(t) | t 2 F } for each root r 2 , * *and the subgroups U = and V = . For each root r, there is a surjection OEr:SL2(F ) i which sends 10t1 to xr(t) and 1t01 to x-* *r(t). This allows the definition of elements hr() = OEr 0 -01 and nr = OEr -0110. T* *he elements_hr(), for r 2 and 2 F *, generate the subgroup H of diagonal elements of G , and together with the nr they generate the subgroup N = of monomial elements. Then N=H ~= W , the Weyl group of G (and of its root system), and B def= = NG(U) is the Borel subgroup of G. Bob Oliver and Yoav Segev * * 63 __ __ __ Now set G = G(F p), and let oe be a Steinberg morphism of G . Set Uoe= CU(o* *e) and Voe= CV (oe), the subgroups of elements fixed by oe, and let bG= * *be the corresponding group of Lie type. Set Hb = CH (oe) \ bG, Nb = CN (oe) \ bG, and* * Bb = CB (oe) \ bG. Let ae be the automorphism of the root system associated to oe,* * as described earlier. In particular ae permutes the positive roots, and hence the* * simple roots. By a root (or simple root) of bGis meant a ae-orbit ^r (or ae-orbit of * *simple roots). Note that if ae = Id, then bGis an (untwisted) Chevalley group, and its* * roots are the roots in the usual sense. We write b = =ae for the set of roots, -^r= {-r |* * r 2 ^r}; and (when J b) for the set of ae-orbits of roots which are linear combinat* *ions of elementsQr 2 ^r2 J. The root subgroup X^rcorresponding to an orbit ^ris the sub* *group ( r2^rXr)oeof oe-invariant elements. The Weyl group of bGis the group cW = bN=* *Hb; or equivalently the subgroup of W = N=H of elements which commute with oe (cf. [Ca* * , Proposition 13.5.2]) when both are considered as groups of permutations of the * *roots (or of the real vector space generated by the roots). The Weyl group is generat* *ed by elements w^sof order two, one for each ae-orbit ^sof simple roots, where the w^* *s-action on sends s to - s for all s 2 ^s. The root subgroups of bGare discussed in detail* * in [Ca , Proposition 13.6.3] and [GLS , Table 2.4]; in particular, they need not be abel* *ian. The Weyl groups of the twisted groups are described in [Ca , x13.3]; each is isomor* *phic to that of some Chevalley group except when bG= 2F4(22k+1), in which case cW is di* *hedral of order 16. For notational convenience we now drop the "hat" from our notation for the fi* *nite simple groups of Lie type of the previous paragraph. Thus from now through the * *end of Appendix D, G = bG, U = bU, etc. Also, we'll abuse notation and write r = ^r* *for a ae-orbit in . Tits has axiomatized the properties of the pairs (B; N) in groups of Lie type* *. These permit, for example, uniform proofs of the simplicity of these groups in all ca* *ses where they are simple. See, e.g., [Ca , x8.2] or [GLS , x1.11] for more detail about * *such BN-pairs. By definition, any group of Lie type is generated by its root subgroups (for * *a given choice of root system). In fact, it suffices to take the simple roots. Lemma D.1. Let G be a finite simple group of Lie type, with root system . The* *n G is generated by the root subgroups Xs and X-s for simple roots s 2 +. Proof.See [Ca , Proposition 13.6.5]. Very briefly, when G is a Chevalley group,* * this holds since conjugation by elements of N (or of W = N=H) permutes the root subgroups * *in the same way as the Weyl group permutes the roots, and each root is in the W -o* *rbit of a simple root. Since N=H is generated by the elements ns 2 for simple* * roots s, this shows that contains all of the Xr for r 2 , and henc* *e_is_all of G. The same argument works for the twisted groups. * *|__| We now turn attention to parabolic subgroups: proper subgroups of G which con* *tain a Borel subgroup. For convenience, set B0 = V H (and B = UH as usual). Let be the root system corresponding to G. For each proper subset J of simple roots of* * G, let be as defined above, and set PJ = = > and PJ0= = >: 64 Fixed point free actions on acyclic 2-complexes By [Ca , Theorem 8.3.2], these are precisely the overgroups of B in G (i.e., th* *e parabolic subgroups containing B). Lemma D.2. Let G be a finite simple group of Lie type. Let be the root syst* *em associated with G, and let + and - be the sets of positive and negative roots. * *Fix a set J of simple roots which does not contain all of them, and let LJ be the s* *ubgroup generated by the diagonal subgroup H together with the root subgroups Xr for all r 2 . Let UJ and VJ be the subgroups generated by all Xr for roots r 2 + or * *r 2 -, respectively, which are not in . Then UJ C PJ = UJLJ and VJ C PJ0= VJLJ, UJ and VJ are nilpotent, and = G. Proof.When G is a Chevalley group, the nilpotency of U UJ and V VJ follows fr* *om [Ca , Theorem 5.3.3], and LJ normalizes UJ and VJ by [Ca , Theorem 8.5.2]. Bot* *h of these are consequences of Chevalley's commutator formula, which says that for a* *ny pair of roots r; s 2 , [Xr; Xs] is generated by the subgroups Xtfor all roots t = ir* *+js where i; j > 0. The twisted group case follows immediately by restriction. And PJ = U* *JLJ and PJ0= VJLJ since U and V are generated by their root subgroups: by definitio* *n when G is a Chevalley group, and by [Ca , Proposition 13.6.1] when G is a twisted gr* *oup. This also shows that = = G. Thus C G, since LJ* * __ normalizes UJ and VJ; and so G = since G is simple. * * |__| The decomposition PJ = UJLJ of Lemma D.2 is called the Levi decomposition of * *PJ, and LJ is called the Levi subgroup. We now return to looking at group actions on 2-dimensional acyclic complexes. Lemma D.3. Let G be a finite simple group of Lie type, and let P $ G be one o* *f the parabolic subgroups PJ or PJ0of Lemma D.2. Then for any action of G on an acycl* *ic 2-complex X, XP 6= ;. Proof.We can assume XG = ;. By Lemma D.2, there are subgroups UJ C PJ, VJ C PJ0, and LJ = PJ \ PJ0, such that UJ and VJ are nilpotent, PJ = UJLJ, PJ0= VJLJ, and = G. In particular, XUJ and XVJ are acyclic, disjoint, and LJ-invarian* *t. Then XLJ 6= ; by Corollary 4.2, applied to the0action of LJ on X with invariant subs* *paces A = XUJ and B = XVJ; and so XPJ and XPJ are nonempty by Lemma 4.3(a). To see this more directly, let Y be the complex obtained by collapsing XUJ an* *d XVJ to separate points. Then Y is still acyclic, LJ acts on Y , and Y LJcontains at* * least the two collapse points. Thus, Y LJis acyclic by Theorem 4.1, is in particular con* *nected, and hence XLJ must intersect0with both subcomplexes XUJ and XVJ. It follows tha* *t_ XPJ = XLJ \ XUJ 6= ; and XPJ = XLJ \ XVJ 6= ;. |__| Appendix E. The four-subgroup criterion In [S1] and [AS ], very strong restrictions were placed on the finite simple * *groups which could possibly have actions on 2-dimensional acyclic complexes without fi* *xed points. The main tool for doing this was a "four subgroup criterion", which fo* *r the sake of completeness we present here as Proposition E.1. To illustrate its use,* * we then Bob Oliver and Yoav Segev * * 65 describe how it was applied to certain multiply transitive groups, and to simpl* *e groups of Lie type and Lie rank at least two _ those cases of the proof of Theorem A w* *hich were not dealt with in Sections 6 and 7. Proposition E.1 [S1, Theorem 3.2].Fix a finite group G and a 2-dimensional acyc* *lic G-complex X. Let H1; H2; H3; H4 G be subgroups such that X6= ; for a* *ny i; j; k. Then X6= ;. Proof.Assume otherwise: that X= ;. Set H = {H1; H2; H3; H4}. By Th* *eo- rem 4.1, XH is the union of the acyclic subcomplexes XHi, which have the proper* *ty that any two or three of them have acyclic intersection, but the four have empty int* *ersection. This implies that H2(XH ) ~=H2(S2) ~=Z (see Lemma 0.1, applied using the poset S of nonempty proper subsets of {1; 2; 3; 4}). But this is impossible, since XH * *_must_be homologically 1-dimensional by Lemma 1.6. |* *__| The simplest application of Proposition E.1 is to multiply transitive groups. Corollary E.2. Assume that G acts 4-transitively on a set S with point stabiliz* *er H G. Let X be a 2-dimensional acyclic G-complex such that XH 6= ;. Then XG 6= * *;. Proof.If |S| = 4, then this follows from Theorem B. So assume |S| 5, and fix f* *our elements s1; s2; s3; s4 2 S. For each i = 1; 2; 3; 4, let Hi G be the subgroup * *of elements which fix sj for all j 6= i. For each {i; j; k; r} = {1; 2; 3; 4}, is the point stabilizer of sr, and hence fixes a point in X by assumption. So XG 6= ; by Pro* *position_ E.1. |__| This is now applied to the alternating groups, as well as most of the Mathieu* * groups. Proposition E.3 [S1, 3.6].If G ~=An for n 6, or if G is one of the Mathieu gro* *ups M11, or M12, then every G-action on an acyclic 2-complex has fixed points. The * *same holds for M23and M24if it holds for M22. Proof.Let X be a 2-dimensional acyclic G-complex. If G = An for n 6, then by Corollary E.2, XG 6= ; if XAn-1 6= ;. By Proposition 6.4 above, A6 ~=L2(9) must* * have nonempty fixed point set, and the result now follows by induction on n. Each of the simple Mathieu groups Mn for n = 11; 12; 23; 24 acts 4-transitive* *ly on a set with point stabilizer Mn-1 (cf. [A3 , 18.9-10 & 19.4], [Gr , 5.33 & 6.18],* * [Mat ], or [Wt ]). So by Corollary E.2, the proposition holds for Mn if it holds for Mn-1.* * Since M10 contains a subgroup A6 of index 2, this proves the proposition when n = 11 or 1* *2; and __ it will follow for the other simple Mathieu groups once it has been shown for M* *22. |__| Proposition E.1 can also be applied to simple groups of Lie type of Lie rank * *at least two. In this case, the subgroups in question come from the root system of the g* *roup. Note that the following proof applies only to groups of Lie type which are them* *selves simple. The Tits group 2F4(2)0, which has index two in 2F4(2), is dealt with h* *ere in Proposition 7.2, as well as in [AS , 5.2]. Proposition E.4 [AS , x5].If G is a simple group of Lie type and Lie rank at le* *ast 2, then every G-action on an acyclic 2-complex has fixed points. 66 Fixed point free actions on acyclic 2-complexes Proof.We use the notation of Lemma D.2. Fix a root system = + q - for G, and let J1q J2 be a decomposition of the set of simple roots as a disjoint union of* * nonempty subsets. For each i = 1; 2, set H+i= and H-i= : The subgroup generated by any three of the Hi is contained in one of the parabo* *lic subgroups PJior PJ0i(in the notation of Lemma D.2), and hence has nonempty fixed point set in X by Lemma D.3. But

= G by Lemma D.1, since it contains * *all __ subgroups Xs and X-s for simple roots s, and hence XG 6= ; by Proposition E.1. * * |__| List of notation: Groups:_ Cm : a cyclic group of order m D2m: a dihedral group of order 2m An: the alternating group on n letters n: the symmetric group on n letters P GLn(q) = GLn(q)=(center): the projective general linear group over Fq Ln(q) = P SLn(q): the projective special linear group over Fq P GUn(q): the projective general unitary group over Fq2 Un(q) = P SUn(q): the projective special unitary group over Fq2 Topological_spaces:_ I = [0; 1]: the unit interval fi Dn = x 2 Rn fikxk 1 : the unit ball in Rn fi Sn = x 2 Rn+1fikxk = 1 : the unit sphere in Rn+1 X ~=Y : X and Y are homeomorphic X ' Y : X and Y are homotopy equivalent X ' *: X is contractible H*(X) def=H*(X; Z) Acyclic_means Z-acyclic: X is acyclic iff H*(X; Z) ~=H*(pt; Z) Families_and_sets_of_subgroups_of_G: S(G): the family of all subgroups of G (H): the conjugacy class of H G F S(G) is a family_() H 2 F implies (H) F SLV (G): the family of solvable subgroups of G MAX (G): the maximal separating family of subgroups of G (G; F) 2 U2 () 9 a 2-dimensional Z-acyclic (G; F)-complex Bob Oliver and Yoav Segev * * 67 For_any_families_F,_F0_of_subgroups_of_G: Fmax: the set of maximal subgroups of F FH = {K 2 F | K H} 8H G F>H = {K 2 F | K % H} 8H G FH= {K 2 F | H $ K $ M} 8H $ M G FH = {K 2 F | K H, some H 2 H} 8H S(G) fi fi F[n]= H 2 F finfi|H| 8n > 1 F^ F0 = {H \ H0| H 2 F; H02 F0} H 2 F is critical_in F () N (F>H ) 6' * Fc = {H 2 F | H critical inF} If_X_is_a_G-complex:_ Gx = {g 2 G | gx = x} 8x 2 X X is a (G; F)-complex () Gx 2 F 8x 2 X XH = {x 2 X | hx = x 8h 2 H}: the fixed point set S X>H = {x 2 X | Gx % H} = K%H XK S XH = H2HXH 8H S(G) S fi X[n]= n||H|XH = {x 2 X fi|Gx 2 nZ} 8n > 1 S -1 X(H)= g2GXgHg S Xs = 16=HGXH = {x 2 X | Gx 6= 1}: the "singular set" of X References [Art]E. Artin, Geometric algebra, Interscience (1957) [A1] M. Aschbacher, Finite group theory, Cambridge Univ. Press (1986) [A2] M. Aschbacher, Overgroups of Sylow subgroups in sporadic groups, Memoirs A* *mer. Math. Soc. 343_(1986) [A3] M. Aschbacher, Sporadic groups, Cambridge Univ. Press (1994) [AS] M. Aschbacher & Y. Segev, A fixed point theorem for groups acting on finit* *e 2-dimensional acyclic simplicial complexes, Proc. London Math. Soc. 67_(1993), 329-354 [Bl] D. Bloom, The subgroups of PSL(3; q) for odd q, Trans. Amer. Math. Soc. 12* *7_(1967), 150-178 [Atl]J. Conway, R. Curtis, S. Norton, R. Parker, & R. Wilson, Atlas of finite g* *roups, Oxford Univ. Press (1985) [Br] G. Bredon, Introduction to compact transformation groups, Academic Press (* *1972) [Ca] R. Carter, Simple groups of Lie type, Wiley (1972) [FR] E. Floyd & R. Richardson, An action of a finite group on an n-cell without* * stationary points, Bull. A.M.S. 65_(1959), 73-76 [Go] D. Gorenstein, Finite groups, Harper & Row (1968) [GLS]D. Gorenstein, R. Lyons, & R. Solomon, The classification of the finite si* *mple groups, nr. 3, Amer. Math. Soc. (1998) [Gra]B. Gray, Homotopy theory, Academic Press (1975) [Gr] R. Griess, Twelve sporadic groups, Springer-Verlag (1998) [Ha] R. Hartley, Determination of the ternary collineation groups whose coeffic* *ients lie in the GF(2n), Annals of Math. 27_(1925), 140-158 [Hu] S.-T. Hu, Homotopy theory, Academic Press (1959) 68 Fixed point free actions on acyclic 2-complexes [H1] B. Huppert, Endliche Gruppen I, Springer-Verlag (1967) [HB2]B. Huppert & N. Blackburn, Finite groups II, Springer-Verlag (1982) [HB3]B. Huppert & N. Blackburn, Finite groups III, Springer-Verlag (1982) [Jac]H. Jacobinski, Genera and decompositions of lattices over orders, Acta mat* *h. 121_(1968), 1-29 [Jan]Z. Janko, A characterization of the smallest group of Ree associated with * *the simple Lie algebra of type (G2), J. Algebra 4_(1966), 293-299 [KS] R. Kirby & M. Scharlemann, Eight faces of the Poincare homology 3-sphere, * *Geometric topology (Proc. Georgia topology conference 1977), Academic Press (1979), 113-146 [Kl1]P. Kleidman, Ph.D. thesis, Imperial College (1987) [Kl2]P. Kleidman, The maximal subgroups of the Chevalley grous G2(q) with q odd* *, the Ree groups 2G2(q) and their automorphism groups, J. Algebra 117_(1988), 30-71 [LW] A. Lundell & S. Weingram, The topology of CW complexes, Van Nostrand (1969) [Mat]E. Mathieu, Sur la fonction cinq fois transitive de 24 quantites, J. Math.* * Pures Appl. 18_(1873), 25-46 [O1] R. Oliver, Fixed point sets of group actions on finite acyclic complexes, * *Comment. Math. Helv. 50_(1975), 155-177 [O2] R. Oliver, Smooth compact group actions on disks, Math. Z. 149_(1976), 79-* *96 [O3] R. Oliver, A proof of the Conner conjecture, Annals of Math. 103_(1976), 6* *37-644 [On] T. Ono, An identification of Suzuki groups with groups of generalized Lie * *type, Annals of Math. 75_(1962), 251-259; Corrigendum, Annals of Math. 77 (1963), 413 [Q1] D. Quillen, Higher algebraic K-theory I, Lecture note in mathematics 341_(* *1973), 85-147 [Q2] D. Quillen, Homotopy properties of posets of nontrivial p-subgroups of a g* *roup, Adv. in Math. 28_(1978), 101-108 [Re] I. Reiner, Maximal orders, Academic Press (1975) [Rim]D. S. Rim, Modules over finite groups, Annals of Math. 69_(1959), 700-712 [S1] Y. Segev, Group actions on finite acyclic simplicial complexes, Israel J. * *Math. 82_(1993), 381-393 [S2] Y. Segev, Some remarks on finite 1-acyclic and collapsible complexes, Jour* *. combinatorial theory 65_(1994), 137-150 [Se] J.-P. Serre, Trees, Springer-Verlag (1980) [St1]R. Steinberg, Lectures on Chevalley groups, Mimeographed notes, Yale Univ.* * (1968) [St2]R. Steinberg, Endomorphisms of linear algebraic groups, Memoirs Amer. Math* *. Soc. 80_(1968) [Sz1]M. Suzuki, On a class of doubly transitive groups, Annals of Math. 75_(196* *2), 105-145 [Sz2]M. Suzuki, Group theory, Springer-Verlag (1982) [Sw] R. Swan, Induced representations and projective modules, Annals of math. 7* *1_(1960), 552-578 [Wh] J. H. C. Whitehead, Combinatorial homotopy I, Bull. Amer. Math. Soc. 55_(1* *949), 213-245 [Wt] E. Witt, Die 5-fach transitiven Gruppen von Mathieu, Abh. Math. Sem. Hansi* *schen Univ. 12_ (1938), 256-264 Bob Oliver Yoav Segev UMR de Mathematiques Department of Mathematics Universite Paris Nord Ben Gurion University 93430 Villetaneuse, France Beer Sheva 84105, Isr* *ael bob@math.univ-paris13.fr yoavs@math.bgu.ac.il