A GEOMETRIC CONSTRUCTION OF SATURATED FUSION SYSTEMS CARLES BROTO, RAN LEVI, AND BOB OLIVER Abstract. A saturated fusion system consists of a finite p-group S, toget* *her with a category which encodes "conjugacy" relations among subgroups of S, and * *which satisfies certain axioms which are motivated by properties of the fusion * *in a Sylow p-subgroup of a finite group. We describe here new ways of constructing * *abstract saturated fusion systems, first as fusion systems of spaces with certain * *properties, and then via certain graphs. A saturated fusion system consists of a finite p-group S, together with a cat* *egory F whose objects are the subgroups of S, whose morphisms are group monomorphisms between those subgroups, and which satisfies certain axioms modelled on the fus* *ion category for the p-subgroups of a finite group. The precise definition of a sa* *turated fusion system is due to Puig [Pu ], and our version of that definition is given* * in Section 1. Saturated fusion systems mimic in several ways the structure of finite group* *s and their classifying spaces. Examples have been known for some time of "exotic" sa* *turated fusion systems _ systems which do not arise from the fusion in any finite group* * _ but the construction of such examples is very complicated, and we are looking for s* *impler and more systematic ways to construct them. One consequence of the main result * *in this paper is a way of constructing a variety of examples of saturated fusion s* *ystems. Of the examples constructed using this technique, some are then shown by other * *means to be exotic. The definition of a fusion system over a p-group S is simple, and in most cas* *es it is clear whether or not a given category satisfies it. In contrast, it is much har* *der to check whether a given fusion system is saturated. For example, for any map f :BS ----* *! X, where S is a finite p-group and X is a topological space, the fusion system of X over (S, f) is a category FS,f(X) whose objects are the subgroups of S, and whe* *re Mor FS,f(X)(P, Q) is the set of all monomorphisms ' 2 Hom (P, Q) such that (f|B* *P ) O B' ' (f|BQ ). This is always a fusion system in the sense of Definition 1.1, bu* *t is not in general saturated. The central result in this paper is Theorem 2.1, where we list some condition* *s on the map f which ensure that the fusion system FS,f(X) is saturated. These condition* *s also ensure that FS,f(X) has an associated linking system (see Definition 1.3), and * *hence that X, S, and f define a p-local finite group. Afterwards, we construct more c* *oncrete examples using that theorem, and show in many cases that they are "exotic" in t* *he sense of not coming from any finite group. For example, in Theorem 4.2, in cer* *tain cases when G is an amalgamated free product of finite groups, we apply our theo* *rem to BG^pto show that the fusion system of G (taken over a maximal p-subgroup of * *G) is ___________ 1991 Mathematics Subject Classification. Primary 55R35. Secondary 55R40, 20D2* *0. Key words and phrases. Classifying space, p-completion, finite groups, fusion. C. Broto is partially supported by MEC grant MTM2004-06686. R. Levi is partially supported by EPSRC grant GR/M7831. B. Oliver is partially supported by UMR 7539 of the CNRS. All of the authors have been supported by the EU grant nr. HPRN-CT-1999-00119. 1 2 CARLES BROTO, RAN LEVI, AND BOB OLIVER saturated. This result, which is stated in terms of trees of groups, was discov* *ered and first proved as a special case of Theorem 2.1, but we also include a more eleme* *ntary, purely graph theoretic proof here. The paper is organized as follows. In Section 1, we give the definitions of a* *bstract fusion and linking systems, as well as definitions of fusion and linking system* *s of groups and spaces and some background results about them. Our main theorem is proven in Section 2. In Section 3, we describe conditions under which the main theorem ca* *n be applied to the space BG^pfor an infinite discrete group G, to prove that the fu* *sion system of G with respect to some finite p-subgroup is saturated (Theorem 3.3). * * A special case of this is then studied in Section 4 _ the case where G acts on a * *tree with finite isotropy subgroups _ and this in turn is applied in Section 5 to co* *nstruct concrete examples of fusion systems, some of which are then shown to be "exotic* *". We hope to find other applications of our main Theorem 2.1 in the future which all* *ow us to construct a still wider variety of examples. We would like to give our thanks to Michael Aschbacher and Andy Chermak, whose construction of the Solomon fusion systems in [AC ] gave us the idea of restati* *ng the results in Section 3 terms of amalgamated free products. 1. A survey of fusion systems We first recall some definitions, mostly from [BLO2 ]. Definition 1.1 ([Pu ] and [BLO2 , Definition 1.1]). A fusion system over a fin* *ite p- group S is a category F, where Ob (F) is the set of all subgroups of S, and whi* *ch satisfies the following two properties for all P, Q S: o Hom S(P, Q) Hom F(P, Q) Inj(P, Q); and o each ' 2 Hom F(P, Q) is the composite of an isomorphism in F followed by an inclusion. Fusion systems as defined above are too general for our purposes, and some ad* *ditional definitions and conditions are needed so that they more closely model the fusio* *n in finite groups. If F is a fusion system over a finite p-subgroup S, then two subgroups * *P, Q S are said to be F-conjugate if they are isomorphic as objects of the category F. Definition 1.2 ([Pu ], see [BLO2 , Definition 1.2]). Let F be a fusion system * *over a p-group S. o A subgroup P S is fully centralized in F if |CS(P )| |CS(P 0)| for all P * *0 S which is F-conjugate to P . o A subgroup P S is fully normalized in F if |NS(P )| |NS(P 0)| for all P 0* * S which is F-conjugate to P . o F is a saturated fusion system if the following two conditions hold: (I)For all P S which is fully normalized in F, P is fully centralized in F * *and Aut S(P ) 2 Sylp(Aut F(P )). (II)If P S and ' 2 Hom F(P, S) are such that 'P is fully centralized, and i* *f we set N' = {g 2 NS(P ) | 'cg'-1 2 AutS('P )}, _ _ then there is ' 2 Hom F(N', S) such that '|P = '. A GEOMETRIC CONSTRUCTION OF SATURATED FUSION SYSTEMS 3 If G is a finite group and S 2 Sylp(G), then the category FS(G) defined in the introduction is a saturated fusion system (see [BLO2 , Proposition 1.3]). An alternative, simplified pair of axioms for a fusion system being saturated* * has been given by Radu Stancu [St]. We now turn to centric linking systems associated to abstract fusion systems.* * When- ever F is a fusion system over a finite p-group S, a subgroup P S is called F* *-centric if CS(P 0) = Z(P 0) for all P 0 S which are F-conjugate to P . We let Fc F d* *enote the full subcategory whose objects are the F-centric subgroups of S. If F = FS(* *G) for some finite group G, then P S is F-centric if and only if P is p-centric in G* *; i.e., if and only if Z(P ) 2 Sylp(CG(P )). Definition 1.3 ([BLO2 , Definition 1.7]). Let F be a fusion system over the p-* *group S. A centric linking system associated to F is a category L whose objects are t* *he F- centric subgroups of S, together with a functor ss :L ------! Fc, and "distingu* *ished" ffiP monomorphisms P --! AutL (P ) for each F-centric subgroup P S, which satisfy the following conditions. (A) ss is the identity on objects. For each pair of objects P, Q 2 Ob (L), Z(P * *) acts freely on Mor L(P, Q) by composition (upon identifying Z(P ) with ffiP(Z(P )) Aut* *L(P )), and ss induces a bijection ~= MorL (P, Q)=Z(P ) ------! Hom F(P, Q). (B) For each F-centric subgroup P S and each x 2 P , ss(ffiP(x)) = cx 2 AutF * *(P ). (C) For each f 2 Mor L(P, Q) and each x 2 P , f OffiP(x) = ffiQ(ssf(x)) Of. A p-local finite group is defined to be a triple (S, F, L), where S is a fini* *te p-group, F is a saturated fusion system over S, and L is a centric linking system associ* *ated to F. The classifying space of the triple (S, F, L) is the p-completed nerve |L|^p. In the following definition, recall that a (possibly infinite) group G is p-p* *erfect if it has no normal subgroup of index p; or equivalently, if Hom (G, Z=p) contains on* *ly the trivial homomorphism. Clearly, if G is generated by p-perfect subgroups, then * *it is itself p-perfect. Hence any group G contains a maximal p-perfect subgroup, whic* *h is normal. Definition 1.4. Fix any pair S G, where G is a (possibly infinite) group and * *S is a finite p-subgroup. (a)Define FS(G) to be the category whose objects are the subgroups of S, and wh* *ere fi -1 MorFS(G)(P, Q) = Hom G(P, Q)def=cg 2 Hom (P, Q) fig 2 G, gP g Q ~= NG(P, Q)=CG(P ). Here cg denotes the homomorphism conjugation by g (x 7! gxg-1), and NG(P, Q)* * = {g 2 G | gP g-1 Q} (the transporter set). (b)For each P S, let C0G(P ) be the maximal p-perfect subgroup of CG(P ). Let* * LcS(G) be the category whose objects are the FS(G)-centric subgroups of S, and where Mor LcS(G)(P, Q) = NG(P, Q)=C0G(P ). Let ss :LcS(G) ___! FS(G) be the functor which is the inclusion on objects * *and sends the class of g 2 NG(P, Q) to conjugation by g. For each FS(G)-centric subgr* *oup 4 CARLES BROTO, RAN LEVI, AND BOB OLIVER P G, let ffiP :P ___! AutLcS(G)(P ) be the monomorphism induced by the in* *clusion P NG(P ). It is clear from the definitions that FS(G) is a fusion system for any S and * *G, and just as clear that it is not always saturated. When G is finite and S 2 Sylp(G)* *, then FS(G) is always saturated (see [Pu ], or [BLO2 , Proposition 1.3]), and LcS(G)* * is a centric linking system associated to FS(G). Thus in this case, (S, FS(G), LcS(G)) is a * *p-local finite group, with classifying space |LcS(G)|^p' BG^p(see [BLO1 , Proposition * *1.1]). When G is infinite, we note the following condition for LcS(G) to be a centri* *c linking system. Lemma 1.5. Fix any pair S G, where G is a (possibly infinite) group and S is a finite p-subgroup, and set F = FS(G). Assume, for each F-centric subgroup P S, that Hi(CG(P )=Z(P ); Fp) = 0 for i = 1, 2. Then LcS(G) is a centric linking s* *ystem associated to F. Proof.Conditions (B) and (C) in Definition 1.3 hold by definition of LcS(G), th* *e pro- jection functor ss, and the distinguished monomorphisms ffiP. Also, for each p* *air of objects P, Q, CG(P ) acts freely on NG(P, Q) by right multiplication, so CG(P )* *=C0G(P ) acts freely on Mor LcS(G)(P, Q) with orbit set Hom F (P, Q). So to prove that * *LcS(G) is a centric linking system associated to F, it remains only to show that for e* *ach F-centric subgroup P S, the inclusion Z(P ) CG(P ) induces an isomorphism Z(P ) ~=CG(P )=C0G(P ). The assumption H1(CG(P )=Z(P ); Fp) = 0 implies that CG(P )=Z(P ) is p-perfec* *t. Since H2(CG(P )=Z(P ); Fp) = 0 and Z(P ) is a finite p-group, the exact sequenc* *es in group cohomology for extensions of modules show that H2(CG(P )=Z(P ); Z(P )) = * *0, and hence that CG(P ) splits as a product Z(P ) x H for a normal subgroup H C CG(P ). Thus H ~=CG(P )=Z(P ) is the maximal p-perfect subgroup of CG(P ), and * *so CG(P )=C0G(P ) = CG(P )=H ~=Z(P ). Fusion systems and linking systems can also be defined for spaces. In the fol* *lowing definition, if H :X x I ---! Y is a homotopy (where I = [0, 1]), then [H] denot* *es its homotopy class among maps X x I ---! Y whose restriction to X x {0, 1} is the s* *ame as that of H. In other words, if we regard H as a path in Map (X, Y ) by adjunc* *tion, then [H] denotes the homotopy class of that path rel endpoints. For any p-group P and any g 2 P , let Hg: BP x I --! BP be the homotopy from IdBP to Bcg induced by the natural transformation of functors B(G) --! B(G) whi* *ch sends the unique object oG in B(G) to the morphism ~gcorresponding to g 2 G. Definition 1.6. Fix a space X, a finite p-group S, and a map f :BS --! X. (a)Define FS,f(X) to be the category whose objects are the subgroups of S, and * *whose morphisms are given by fi Hom FS,f(X)(P, Q) = ' 2 Inj(P, Q) fif|BP ' f|BQ OB' for each P, Q S. (b)Define F0S,f(X) FS,f(X) to be the subcategory with the same objects as FS,* *f(X), and where MorF0S,f(X)(P, Q) (for P, Q S) is the set of all composites of r* *estrictions of morphisms in FS,f(X) between FS,f(X)-centric subgroups. A GEOMETRIC CONSTRUCTION OF SATURATED FUSION SYSTEMS 5 (c)Define LcS,f(X) to be the category whose objects are the FS,f(X)-centric sub* *groups of S, and whose morphisms are defined by fi Mor LcS,f(X)(P, Q) = (', [H]) fi' 2 Inj(P, Q), H :BP x I ---! X, H|BPx0 = f|BP , H|BPx1 = f|BQ OB' . The composite in LcS,f(X) of morphisms (',[H]) (_,[K]) P ------! Q ------! R , where H :BP x I ! X and K :BQ x I ! X are homotopies as described above, are defined by setting (_, [K]) O(', [H]) = (_ O', [(K O(B' x Id)) . H]), where .denotes composition (juxtaposition) of homotopies. Let ss :LcS,f(X) ------! FS,f(X) be the forgetful functor: it is the inclusion on objects, and sends a morphi* *sm (', [H]) to '. For each FS,f(X)-centric subgroup P S, let ffiP :P ------! AutLcS,f(X)(P ) be the "distinguished homomorphism" which sends g 2 P to (cg, [f|BP OHg]). Equivalently, via adjunction, a morphism from P to Q in LcS,f(X) can be thoug* *ht of as a pair (', [H]), where ' 2 Hom (P, Q), H is a path in the mapping space Map * *(BP, X) from f|BP to f|BQ OB', and [H] is the homotopy class of the path H rel endpoint* *s. The categories F0S,f(X) FS,f(X) are always fusion systems over S, but are n* *ot in general saturated. However, in certain situations we consider, F0S,f(X) wil* *l be a saturated fusion system, even though FS,f(X) might not be (see Example 3.4). Theorem A of [BCGLO1 ] says that if all morphisms in a fusion system are ob* *tained as composites of restrictions of morphisms between centric subgroups, then it is s* *aturated if the saturation conditions (I) and (II) hold on centric subgroups. Thus it ma* *kes sense, for a general abstract fusion system F, to define the subsystem F0 F over the* * same p-group S to be the subcategory with the same objects, but with only those morp* *hisms which are obtained as composites of restrictions of morphisms in F between F-ce* *ntric subgroups. One particularly well behaved situation is that in which F0 has no * *more centric subgroups than those already centric in F. In this case, it clearly fol* *lows that the full subcategories of centric objects in F and in F0 are equal, and hence t* *hat one can check conditions (I) and (II) in either subcategory. These arguments are collected in the following proposition. Proposition 1.7. Fix a space X, a finite p-group S, and a map f :BS ---! X. If * *all FS,f(X)-centric subgroups P S satisfy conditions (I) and (II) in Definition 1* *.2, and if all F0S,f(X)-centric subgroups of S are FS,f(X)-centric, then F0S,f(X) is a * *saturated fusion system. In the situation of Proposition 1.7, there could possibly be a F0S,f(X)-centr* *ic sub- group P S which is not FS,f(X)-centric, because it is FS,f(X)-conjugate to a * *sub- group which is not centric in S. When this is the case, [BCGLO1 , Theorem 2.2* *] cannot be applied to prove the above proposition, since we've changed the set of centr* *ic sub- groups in question. This is why we need to assume that the two fusion systems h* *ave the same centric subgroups. 6 CARLES BROTO, RAN LEVI, AND BOB OLIVER f Our main theorem will give some conditions on a map BS ----! X which ensure that a triple (S, F0S,f(X), LcS,f(X)) is a p-local finite group. More generally* *, however, without any extra hypotheses, the category LcS,f(X) does satisfy most of the ax* *ioms for being a centric linking system associated to FS,f(X). In the following lemma, for any f :BS ---! X as above, and any P S, we let !P :BZ(P ) ------! Map (BP, X)f|BP be the map which is adjoint to the composite B~ f BZ(P ) x BP -----! BS -----! X, where ~: Z(P ) x P ---! S is multiplication. Lemma 1.8. Fix a space X, a finite p-group S, and a map f :BS ---! X. Then the category LcS,f(X), together with the functor ss :LcS,f(X) ------! FS,f(X) and the distinguished homomorphisms ffiP :P ---! AutLcS,f(X)(P ), satisfy axio* *ms (B) and (C) in Definition 1.3. If in addition, ss1(!P) Z(P ) ____! ss1 Map (BP, X), f|BP is an isomorph. 8 FS,f(X)-centric P (*S,) then LcS,f(X) is a linking system associated to FS,f(X). Proof.Proving this means essentially repeating the proof of [BLO2 , Theorem 7.* *5]. Set F = FS,f(X) and L = LcS,f(X) for short. Condition (B) in Definition 1.3 clearly* * holds. For g 2 P S, set Hbg = f|BP O Hg: a homotopy BP x I ! X from f|BP to ffiP f|BP O Bcg. Thus the distinguished homomorphism P ----! AutL(P ) is defined by sending g 2 P to (cg, [Hbg]). Condition (C) means showing, for each (', [H]) 2 Mor L(P, Q) and each g 2 P ,* * that the following square commutes: (',[H]) P ___________! Q | | (cg,[Hbg])| (c'(g),[Hb'(g)])| # # (',[H]) P __________! Q. Here, H :BP x I ---! X is a homotopy from f|BP to f|BQ OB'. Clearly, ' Ocg = c'(g)O'. It remains to check that the two juxtaposed homotopies described in t* *he following diagram are homotopic among homotopies from f|BP to f|BQ OB(' Ocg): H f|BP ______________f|BQ OB' Hbg|| Hb'(g)O(B'xId)|| | HO(BcgxId) | f|BP OBcg _________ f|BQ OB(' Ocg). The map F :BP x I x I ----! X defined by F (x, s, t) = H Hg(x, t), s A GEOMETRIC CONSTRUCTION OF SATURATED FUSION SYSTEMS 7 defines a homotopy between them, since F (x, s,=0)H(x, s), F (x, 1,=t)(f|BQ OB' OHg)(x, t) = bH'(g)(B'(x), t), F (x, 0,=t)f|BP OHg(x, t) = bHg(x, t), and F (x, s,=1)H(Bcg(x), s). It remains to prove (A) while assuming that (*) holds. For any F-centric subg* *roup P S, we identify ss1(Map (BP, X), f|BP ) as a subgroup of AutL(P ): the subgr* *oup of elements of the form (Id, [H]) when H is a homotopy from f|BP to itself. Under * *this identification, ffiP restricts to the homomorphism from Z(P ) to ss1(Map (BP, X* *), f|BP ) which sends g 2 Z(P ) to [Hbg], where bHgis now regarded as a loop in Map (BP, * *X). By definition of F and L, for any other F-centric subgroup Q S, ss1(Map (BP, X),* * f|BP ) acts freely on Mor L(P, Q) with orbit set Hom F(P, Q). So to prove (A), we must* * show that the isomorphism ss1(!P) of (*) sends g 2 Z(P ) to [Hbg]. Let [1] be the category with two objects 0, 1, and one nonidentity morphism 0* * ! 1. Fix g 2 Z(P ), and consider the composite functor Idx_ B(~) : B(P ) x [1] ------! B(P ) x B(Z(P )) ------! B(P ), where _ :[1] ___! B(Z(P )) sends 0 ! 1 to the morphism g, and where B(~) is in* *duced by multiplication. Then | |: BP x I -----! BP is induced by the natural homomorphism of functors from IdB(P)to itself defined* * by sending the object oP to the morphism corresponding to g, and is thus the homot* *opy Hg of Definition 1.6. By definition, ss1(!P)(g) is the homotopy class of f O| |* *, when regarded as a loop in Map (BP, X), and is thus equal to [f OHg] = [Hbg]. This f* *inishes the proof of (A), and hence of the lemma. We will refer several times to the following classical result. Proposition 1.9. For any pair of discrete groups H and G, the natural map Rep (H, G) def=Hom(H, G)= Inn(G) ------! [BH, BG] is a bijection. For each ae 2 Hom (H, G), the homomorphism CG(ae(H)) x H ___! * *G is adjoint to a homotopy equivalence BCG(ae(H)) ---'---!Map (BH, BG)Bae. Proof.See, for example, [BrK , Proposition 7.1]. 2. A new topological characterization of fusion systems In this section, we show, for a p-complete space X, a p-group S, and a map f * *:BS ! X, that the triple (S, F0S,f(X), LcS,f(X)) is a p-local finite group if X, S, a* *nd f satisfy certain conditions listed in Theorem 2.1 below. When S is a p-group, a map f :BS --! X will be called Sylow if every map BP --! X, for a p-group P , factors through f up to homotopy. A map f :X --! Y between arbitrary spaces is called centric if the induced map fO- Map (X, X)Id------! Map (X, Y )f 8 CARLES BROTO, RAN LEVI, AND BOB OLIVER is a homotopy equivalence. In [BLO2 , Theorem 7.5], we showed that a p-complete space X is the classify* *ing space of some p-local finite group if and only if there is a pair (S, f), where S is * *a p-group and f :BS ! X is a map, such that (a) FS,f(X) is saturated, (b) X ' |LcS,f(X)|^p, a* *nd (c) f|BP is a centric map for each FS,f(X)-centric subgroup P S. The following th* *eorem is similar in nature, although aimed at finding conditions for (S, FS,f(X), LcS* *,f(X)) to be a p-local finite group rather than for X to be the classifying space of a p-* *local finite group. The main new result here is the geometric condition for the fusion syst* *em FS,f(X) to be saturated. Theorem 2.1. Fix a space X, a p-group S, and a map f :BS --! X. Assume that (a)f is Sylow; (b)f|BP is a centric map for each FS,f(X)-centric subgroup P S; and (c)every F0S,f(X)-centric subgroup of S is also FS,f(X)-centric. Then the triple S, F0S,f(X), LcS,f(X) is a p-local finite group. Proof.For each FS,f(X)-centric subgroup P S, (b) implies that composition with f|BP induces a homotopy equivalence Map (BP, BP )Id___! Map (BP, X)f. Also, by Proposition 1.8, Map (BP, BP )Id ' BZ(P ), and the resulting homotopy equivalen* *ce j BZ(P ) ___'! Map (BP, X)f is adjoint to the composite B~ f BZ(P ) x BP -----! BS -----! X, where ~: Z(P ) x P ---! S is multiplication. Thus !P = j, where !P is the map * *of Lemma 1.8, and hence is a homotopy equivalence. Condition (*) of Lemma 1.8 thus holds, and so LcS,f(X) is a linking system as* *sociated to FS,f(X) by that lemma. It remains only to prove that F0S,f(X) is saturated. * *This proof is based on two lemmas which will be stated and proven later in this sect* *ion. Write L = LcS,f(X), F = FS,f(X), and F0 = F0S,f(X) for short. By Proposition 1.7 and (c), in order to prove that F0 is saturated, it suffices to show that c* *onditions (I) and (II) in Definition 1.2 hold for all F0-centric subgroups P S. If P * * S is F0-centric, then it is also F-centric by (c), and hence f|BP is a centric map b* *y (b). We first prove condition (I). Assume P S is F0-centric and fully normalized* * in F0. Since P is F0-centric, it is fully centralized, and it remains only to sho* *w that AutS(P ) 2 Sylp(Aut F0(P )). We identify P with ffiP(P ) Aut L(P ). Since L* * is a centric linking system associated to F or F0, the homomorphism ssP,P: AutL(P ) ------! Aut F(P ) = AutF0(P ) induced by the functor ss :L ___! F is surjective with kernel Z(P ). Also, ssP* *,P(P ) = Inn(P ) is normal in Aut F(P ), and thus P C Aut L(P ). By axiom (B) for a lin* *king system, ssP,P sends g 2 AutL(P ) to cg 2 Aut(P ), and thus CAutL(P)(P ) = Ker(ssP,P) = Z(P ). (1) By Lemma 2.2, f|BP extends up to homotopy to a map bf:B AutL(P ) --! X (by definition, AutL(P ) = AutLcP,f(X)(P )). If T is any Sylow p-subgroup of AutL(P* * ), then T P since P C Aut L(P ), fb|BT factors through BS by condition (a), and t* *hus there is a homomorphism ': T --! S such that '|P 2 Hom F0(P, S). In particula* *r, A GEOMETRIC CONSTRUCTION OF SATURATED FUSION SYSTEMS 9 '|P is a monomorphism. Hence since Z(T ) CT(P ) P by (1), Ker(') \ Z(T ) Ker(') \ P = 1; and (since a nontrivial normal subgroup intersects nontrivially* * with the center) this implies that ' is a monomorphism. Hence |NS('(P ))| |T | si* *nce NS('(P )) '(T ); and also |NS(P )| |NS('(P ))| since P is fully normalized.* * Since P is centric in S, | AutS(P )| = |NS(P )|=|Z(P )| |NS('(P ))|=|Z(P )| |T=Z(P )| ; and thus Aut S(P ) 2 Sylp(Aut F0(P )) since Aut F0(P ) ~=Aut L(P )=Z(P ) and* * T 2 Sylp(Aut L(P )). It remains to prove condition (II). Fix a morphism ' 2 Hom F0(P, S), and set N = N' = {g 2 NS(P ) | 'cg'-1 2 AutS('(P ))} as usual. Consider the diagram B' BP _______//_BS99 B'0 r r incl|| rr f|| fflffl|f|BNfflffl|r BN ________//X . The square commutes up to homotopy since ' is a morphism in F0, and condition (* *1) in Lemma 2.3 holds by definition of N. Thus, by Lemma 2.3, there is a homomorph* *ism '02 Hom (N, S) such that B'0makes both triangles in the above diagram commute up to homotopy. The commutativity of the lower triangle means that '02 Hom F0(N, S* *). The commutativity of the upper triangle implies that '0|P = ' Ocg for some g 2 P _ def0 -1 (Proposition 1.9), and thus ' = ' Ocg is an extension of ' which lies in Hom F* *0(N, S). This finishes the proof of (II). It remains to state and prove the technical lemmas used in the proof of Theor* *em 2.1. Lemma 2.2. Fix a space X, a p-group P , and a centric map f :BP --! X. Set L = LcP,f(X) for short. Then f extends (up to homotopy) to a map _ f :B AutL(P ) ------! X. Proof.We first consider the following abstract situation. Fix a space Y , a bas* *epoint y0 2 Y , and a finite group G with a right action on Y . Consider the following* * commu- tative diagram '1 (Y xG EG) ______!wG _______! Y x EG ___! Y xG EG | ww | pr1'| ww |' # # '2 F ___________! G __________!Y . Here, '1 and '2 are defined by the action at the basepoints: '1(g) = (y0g, g-1* *) and '2(g) = y0g. Also, F is the "standard" homotopy fiber of '2: fi F = (g, H) fig 2 G, H :I ! Y, H(0) = y0, H(1) = y0g . (I = [0, 1]) This is an H-space, via the product (g0, H0)(g, H) = (g0g, (Rg OH0) . H), where* * Rg denotes the right action of g on Y and "." denotes composition of paths in Y . * *We can also regard (Y xG EG) as the standard homotopy fiber of '1. Then pr1 is defin* *ed 10 CARLES BROTO, RAN LEVI, AND BOB OLIVER by projecting a path in Y x EG to the first factor, and is a map of H-spaces an* *d a homotopy equivalence. In particular, it induces an isomorphism of groups ss0(pr1) ss1(Y xG EG) ~=ss0( (Y xG EG)) ------! ss0(F ) . (1) Set F = FP,f(X) for short. We apply the above remarks to the space Y = Map (BP, X)f, the point y0 = f, and the group G = Aut F(P ), where ff 2 Aut F(P* * ) acts on Y via right composition by Bff. Thus after replacing paths in Y by homo* *topies, fi F = (', H) fi' 2 AutF (P ), H :BP x I ---! X, H|BPx0 = f, H|BPx1 = f OB' , and AutL(P ) = ss0(F ) by definition. Also, since f is centric, Y = Map (BP, X)f ' Map (BP, BP )Id' BZ(P ), where the last equivalence follows from Proposition 1.9. Then Y xG EG is also a* *spher- ical, and so Y xG EG ' B AutL(P ) by (1). Since B' fixes the base point of BP for all ' 2 Aut(P ), the evaluation map Y = Map (BP, X)f --eval----!X is AutF(P )-equivariant (with respect to the trivial action on X). It thus fact* *ors through the orbit space, or alternatively through the Borel construction: _ eval f: B AutL(P ) ' Map (BP, X)f xAutF(P)E AutF(P ) -------! X . _ It remains to show that f|BP ' f, where BP is included into B AutL(P ) via the distinguished monomorphism ffiP. By the naturality of these maps, it_suffices t* *o do this when X = BP and f = Id. In this case, that means showing that ss1(f|BP ) = IdP.* * Fix g 2 P , and let Hg: BP x I ---! BP be as in Definition 1.6. We also regard Hg a* *s a path in Map (BP, BP )Idfrom IdBP to Bcg, whose restriction to the basepoint of * *BP is by definition the loop in BP representing g. By the above construction, g 2 ss1* *(BP ) corresponds to the class [Hg, OE] 2 ss1 Map (BP, X)f xAutF(P)E AutF(P ) , where OE is any path in EP E AutF(P ) from the vertex Idto the vertex c-1g. H* *ence upon evaluating this at the basepoint_of BP , we see that eval([Hg, OE]) is the* * loop in BP representing g, and thus that ss1(f)(g) = g. It remains to prove the existence of certain homotopy liftings. Lemma 2.3. Fix a finite group H, a normal p-subgroup P C H, a p-group S, and a monomorphism ': P --! S such that CS('(P )) = Z('(P )). Let X be a space, and let f :BS --! X be such that f OB' is centric. Assume that for each x 2 H, 'cx'-1 2 AutS('(P )). (1) Let s: BH --! X be such that the square in the following diagram commutes up to homotopy: B' BP _______//_BS99 B'0 r r incl|| rr f|| (2) fflffl|sr fflffl| BH ________//X . Then there is a homomorphism '02 Hom (H, S) such that the two triangles in diag* *ram (2) commute up to homotopy. A GEOMETRIC CONSTRUCTION OF SATURATED FUSION SYSTEMS 11 Proof.We identify BP with EH=P and BH with ffi E(H=P ) x EH H = E(H=P ) xH=P EH=P. Thus the inclusion BP BH is induced by the inclusion of an orbit H=P E(H=P * *). Let b: EH=P -----! BS and bs:E(H=P ) xH=P EH=P -----! X be maps homotopic to B' and s under these identifications. By (1), the connected component Map (EH=P, BS)B' is invariant under the action of H=P induced by the action of the group on EH=P . We thus get the following s* *quare of equivariant maps between spaces with (H=P )-action H=P ___v_____//Map(EH=P,5BS)b5 kk k incl|| keukk ' |fO-| (3) fflffl|ukk fflffl| E(H=P ) _______//Map(EH=P, X)fOb. Here, u is adjoint to bs(when regarded as a map defined on E(H=P ) x EH=P ); an* *d v is defined by setting v(gP )(xP ) = b(xgP ) for x 2 EH. The square in (3) commu* *tes up to equivariant homotopy by the commutativity of the square in (2). Now, Map (EH=P, BS)B' ' BCS('(P )) ' BZ(P ) by Proposition 1.9, and since '(P ) is centric in S by assumption. Also, Map (EH=P, X)fOb ' BZ(P ) since f OB* *' is a centric map by assumption. Thus the map (f O-) is H=P -equivariant and a homo* *topy equivalence. Since the H=P -action on E(H=P ) is free, there is an equivariant * *lifting euof u as in the above diagram which makes both triangles in (3) commute up to equiv* *ariant homotopy. This is adjoint to an H=P -equivariant map from E(H=P ) x EH=P to BS, which (since H=P acts trivially on BS) factors through es:BH = E(H=P ) xH=P EH=P ------! BS which makes the two triangles in (2) commute up to homotopy. Finally, es' B'0for some '02 Hom (H, S) by Proposition 1.9 again, and this finishes the proof. 3.Fusion systems of completed classifying spaces of groups In order to apply Theorem 2.1 to a space X, we must have good control over the mapping spaces Map (BP, X) for finite p-groups P . One interesting case where w* *e can do this is when X = BG^pfor certain infinite groups G. This is based on a theor* *em of Broto and Kitchloo [BrK ]. When G is an infinite group, we say that a subgroup S G is a Sylow p-subgro* *up if S is a finite p-subgroup, and if all other finite p-subgroups of G are conju* *gate to subgroups of S. Proposition 3.1. Fix a prime p and a discrete group G. Assume there is an Fp-ac* *yclic G-complex X with finitely many orbits of cells and with finite isotropy subgrou* *ps. Let S G be any finite p-subgroup, and let f :BS ---! BG^pbe the inclusion. Then t* *he following hold. (a)FS,f(BG^p) = FS(G). (b)If S is a Sylow p-subgroup of G, then the map f is Sylow. 12 CARLES BROTO, RAN LEVI, AND BOB OLIVER (c)For any P S, f|BP is a centric map if and only if Hi(CG(P )=Z(P ); Fp) = 0* * for all i > 0. Proof.In the notation of [BrK ], K1X is a class of topological groups which inc* *ludes all discrete groups which act on Fp-acyclic complexes with finitely many orbits* * of cells and with finite isotropy subgroups. (The definition in [BrK ] also requires tha* *t the fixed point set of any finite p-group be Fp-acyclic, but this follows from Smith theo* *ry, since the complex is finite dimensional.) In particular, this class includes G. Hence* * by [BrK , Corollary 3.3], for any finite p-group P , the natural map ~= ^ Rep (P, G) def=Hom(P, G)= Inn(G) ------! [BP, BGp] (1) * *(ae,incl) is a bijection. Also, for each ae 2 Hom (P, G), the homomorphism P xCG(ae(P )) * *----! G induces a homotopy equivalence BCG(P )^p---'---!Map (BP, BG^p)Bae. (2) Point (a) follows immediately from (1). Assume S is a Sylow p-subgroup of G. If P is any finite p-group and s: BP ! B* *G^p is a map, then s ' B' for some ' 2 Hom (P, G) by (1), '(P ) is G-conjugate to s* *ome Q S since S is Sylow, and thus B' ' f OB'0for some '02 Hom (P, S). Thus the map f is Sylow, and this proves (b). By (2), for any P S, f|BP is a centric map if and only if the inclusion of * *BZ(P ) into BCG(P )^pis a homotopy equivalence, or equivalently, if the inclusion of B* *Z(P ) into BCG(P ) is an Fp-homology isomorphism. Since Z(P ) is central in CG(P ), * *this last condition is equivalent to requiring that Hi(CG(P )=Z(P ); Fp) = 0 for all* * i > 0, and this proves (c). Before stating our theorem, we need one more definition. Definition 3.2. Fix a prime p. fi (a)If H G are finite groups, then H is strongly embedded in G at p if pfi|H|,* * but H \ gHg-1 has order prime to p for all g 2 Gr NG(H). (b)If G is a finite group and S 2 Sylp(G), a subgroup P S is essential if eit* *her P = S, or P is p-centric in G and Out G(P ) has a strongly embedded subgroup* * at p. By Goldschmidt's version of Alperin's fusion theorem [Gd , Theorem 3.3], for * *any finite group G and any S 2 Sylp(G), each morphism in FS(G) is a composite of restrictions of morphisms between subgroups of S which are essential in G. Note* * that each essential subgroup is also radical _ OutG (P ) has no strongly embedded su* *bgroup if Op(Out G(P )) 6= 1. A finite group G has a strongly embedded subgroup H if and only if the poset * *of nontrivial p-subgroups of G is disconnected (cf. [As , 46.6]), in which case th* *e stabilizer of a connected component is strongly embedded. The following theorem is a first application of Theorem 2.1. Theorem 3.3. Fix a prime p and a discrete group G. Let X be an Fp-acyclic G- complex with finitely many orbits of cells and with finite isotropy subgroups. * * Fix a vertex x* 2 X, let G* be the isotropy subgroup of x*, choose S 2 Sylp(G*), and * *set F = FS(G) and L = LcS(G). Assume the following hold: A GEOMETRIC CONSTRUCTION OF SATURATED FUSION SYSTEMS 13 (a)For each finite p-subgroup P G, XP contains at least one point in the orbi* *t Gx*. (b)If P S is F-centric, then XP =CG(P ) is Fp-acyclic. (c)If P S is a Sylow p-subgroup of the isotropy subgroup of an edge of X, or * *an essential p-subgroup of the isotropy subgroup of a vertex, then P is F-centr* *ic. Then F is a saturated fusion system over S, and L is a centric linking system a* *ssociated to F. Proof.If f :BS ---! BG^pis the map induced by the inclusion, then F = FS,f(BG^p) by Proposition 3.1(a). For any finite p-subgroup P G, there is g 2 G such th* *at gx* 2 XP by point (a) above, and hence P is contained in the isotropy subgroup gG*g-1 of gx*. Since gSg-1 2 Sylp(gG*g-1), this shows that P is G-conjugate to* * a subgroup of S. Thus S is a Sylow p-subgroup of G; and hence by Proposition 3.1(* *b), the map f is Sylow. By Theorem 2.1, to prove that F = FS(G) is saturated, it remains only to check condition 2.1(b), and to show that F = F0S,f(BG^p). (This last claim also impl* *ies condition 2.1(c).) In Step 1, we prove condition 2.1(b), and also prove that L * *= LcS(G) is a centric linking system associated to F. In Step 2, we prove that F = F0S,f* *(BG^p); i.e., that F is generated by morphisms between F-centric subgroups. Step 1: Fix an F-centric subgroup P S. Thus CS(P 0) = Z(P 0) for each P 0 S which is G-conjugate to P . If Q CG(P ) is a finite p-subgroup, then P Q is a* * finite p-group, so gP Qg-1 S for some g 2 G, and Q Z(P ) by the above remark appli* *ed to P 0= gP g-1. Thus Z(P ) is maximal among finite p-subgroups of CG(P ). __ Set C G(P ) = CG(P )=Z(P ) for short. For each x 2 XP , Gx is a finite group * *which contains P , so CGx(P )=Z(P ) is finite of order prime to p, and hence its clas* *sifying space is Fp-acyclic. Consider the projection maps __ pr1 __ pr2 __ BC G(P ) ------ EC G(P ) x_CG(P)XP ------! XP =C G(P ) = XP =CG(P ) associated to the Borel construction on XP . All fibers (point inverses) of pr* *1 are homeomorphic to XP , and thus Fp-acyclic by Smith theory (X is Fp-acyclic and f* *inite dimensional). For each x 2 XP with orbit ~x2 XP =CG(P ) and with stabilizer sub* *group Gx, __ pr-12(~x) ~=EC G(P )=CGx(P ) ' B(CGx(P )=Z(P )) is Fp-acyclic. Hence by a spectral sequence argument (or by an appropriate vers* *ion of the Vietoris mapping theorem), pr1and pr2are both Fp-homology_equivalences. Sin* *ce XP =CG(P ) is Fp-acyclic by assumption, this implies that BC G(P ) is Fp-acycli* *c. __ Thus Hi(C G(P ); Fp) = 0 for all i > 0. Hence by Proposition 3.1(c), the map * *f|BP is centric, and this finishes the proof of condition 2.1(b). By Lemma 1.5, this al* *so shows that L = LcS(G) is a centric linking system associated to F. Step 2: Fix any ' = cg 2 Hom F(P, Q) in F. Then P S and gP g-1 S, so P is contained in the isotropy subgroups of both x* and g-1(x*). Choose a path OE in* * the 1- skeleton of XP from x* to g-1(x*). Let v0 = x*, v1, . .,.vm = g-1(x*) be the su* *ccessive vertices in the path OE, let ei be the edge connecting vi-1to vi, and set Hi = * *Geiand Ki = Gvi. Thus by construction, P Hi, and Ki-1 Hi Ki for all 1 i m. Also, S 2 Sylp(K0) and Km = g-1K0g. 14 CARLES BROTO, RAN LEVI, AND BOB OLIVER Fix Sylow subgroups Pi2 Sylp(Hi) such that P Pi. Choose Qi, Q0i2 Sylp(Ki) s* *uch that Q0i-1 Pi Qi, and let ki2 Kibe such that Q0i= kiQik-1i. We also assume th* *at Q0 = S and Q0m= g-1Sg. Finally, since S is Sylow in G, there are elements gi 2 G such that Qi giSg-1i. In particular, when i = m, since g-1Sg = Q0m= km Qm k-1m, we can choose gm = k-1mg-1. Consider the following diagram, where all subgroups are contained in S: _0 k0 '1 g1 _1 k1g1'2 g2 km-1gm-1'm gm _m kmgm P ___! P __! P __! P __! P P __! P __! P | | | | | | #| | | | . . . | | _'1 # # _'2 # # _'m # P1k0__! P1g1 P2k1g1_! P2g2 Pmkm-1gm-1_! Pmgm. Here, we use the standard notation_Hg = g-1Hg. Also, _iis conjugation by g-1ik-* *1igi2 Kgii(where g0 = 1), and 'i and ' iare conjugation by g-1iki-1gi-1. All of these subgroups are contained in S by construction. Also, by the above choice of gm ,* * _m is conjugation by ggm . Thus the composite of these morphisms _i and 'i is conjuga* *tion by (ggm )(g-1mkm-1 gm-1 )(g-1m-1k-1m-1gm-1 ) . .(.g-12k1g1)(g-11k-11g1)(g-11k0g0)(* *g-10k-10g0) = g. (Recall that g0 = 1.) By (c), each subgroup of S which is conjugate to any Pi is F-centric. Each _i is a morphism in the fusion system of Kgii, and hence a composite of restrictio* *ns of morphisms between essential p-subgroups of this group [Gd , Theorem 3.3]. Since* * all such subgroups are F-centric by (c), this finishes the proof. We finish the section with two very simple examples which illustrate why some* * of these assumptions are needed. The first example shows why condition (c) is need* *ed in Theorem 3.3. It also shows why we cannot take F0 = F in Theorem 2.1. Example 3.4. Let S be an abelian p-group, T S a proper subgroup of order > 2, and H Aut (T ) a nontrivial subgroup of order prime to p. Set G = S *(T oH). T Then G acts on a tree with isotropy subgroups all G-conjugate to S, T , or T oH* *. This action satisfies conditions (a) and (b) in Theorem 3.3, but not condition (c); * *and the inclusion map f :BS ---! BG^psatisfies all of the hypotheses (a)-(c) in Theorem* * 2.1. The fusion system FS(G) = FS,f(BG^p) is not saturated. The fusion system F0S,f(* *BG^p) is equal to FS(S), and is thus a proper subsystem of FS(G) and is saturated. Proof.By [Se, Theorem I.9], G acts freely on a tree X with isotropy subgroups c* *onju- gate to S and T oH on vertices, and to T on edges, and with fundamental domain * *an interval. Since S does not fix any edges (and XS must be a tree), XS is a point* *, and hence XS=CG(S) is also a point. Since T has index prime to p in T oH, XP contai* *ns elements in the orbit G=S for each finite p-subgroup P G. Thus the action of * *G on X satisfies conditions 3.3(a) and 3.3(b). Conditions (a)-(c) in Theorem 2.1 the* *n follow using Proposition 3.1. The fusion system F = FS(G) is not saturated, since the automorphisms in the group AutF (T ) ~=H do not extend to automorphisms in AutF (S). A GEOMETRIC CONSTRUCTION OF SATURATED FUSION SYSTEMS 15 The next example shows that condition (b) in Theorem 2.1 and condition (b) in Theorem 3.3 must be assumed (in each theorem) for all F-centric subgroups: it d* *oes not suffice to assume them when P = S. Example 3.5. Set p = 2, S = D8 x C22, T = C4 x C22, and H = C4 x A4, with the obvious inclusions of T in S and H. Set G = S * H. Then G acts on a tree with T all isotropy subgroups G-conjugate to S, T , or H. This action satisfies condit* *ions (a) and (c) in Theorem 3.3 as well as conditions (a) and (c) in Theorem 2.1, but do* *es not satisfy condition (b) in either theorem. The inclusion map f :BS ---! BG^pis Sy* *low and centric, but the inclusion map f|BT :BT --! BG^pis not centric. Neither fu* *sion system FS(G) = FS,f(BG^p) nor F0S,f(BG^p) is saturated. Proof.Set F = FS,f(BG^p) and F0 = F0S,f(BG^p) for short. As in the last example* *, G acts on a tree X with isotropy subgroups as described, and with fundamental dom* *ain an interval, by [Se, Theorem I.9]. Since any action of a finite group on a tre* *e has a fixed point, every finite subgroup of G is contained in an isotropy subgroup, a* *nd thus in a subgroup G-conjugate to S or H. This shows that S is a Sylow 2-subgroup of* * G, and hence (by Proposition 3.1(b)) that f is Sylow. Thus both conditions 2.1(a) * *and 3.3(a) hold. Also, f is centric by Proposition 3.1(c), since CG(S) = Z(S). Since S is a 2-group and H has a normal Sylow 2-subgroup, their only radical * *2- subgroups (hence their only essential 2-subgroups) are S and T , respectively. * * Since both are centric in S, condition 3.3(c) holds. As seen in Step 2 of the proof o* *f Theorem 3.3, this implies that F0 = F, and thus that condition 2.1(c) also holds. Now, T is normal in G, since it is normal in S and H, and G=T ~=(S=T ) *(H=T * *) ~= C2* C3. Hence CG(T )=T ~=Ker [C2* C3 ---! C2 x C3], and this is a free group (since it acts freely on a tree). In particular, H1(CG* *(T )=T ; F2) 6= 0; and (since CG(T )=T acts freely on the tree XT ) XT =CG(T ) ' B(CG(T )=T ) i* *s not F2-acyclic. So by Proposition 3.1(c), f|BT is not a centric map; and this shows* * that conditions 2.1(b) and 3.3(b) both fail. Now, Aut F(C4 x C22) ~=C2 x C3, but Out F(D8 x C22) ~=1 (and the same for F0). The automorphism of C4 x C22of order 3 thus fails to extend to S, so axiom (II)* * fails, and F is not saturated. 4. Fusion systems of trees of groups Let (G, T ) be a tree of groups in the sense of [Se, xI.4.4]. Thus T is a t* *ree; and G assigns groups G(v) and G(e) to each vertex v 2 T 0and each edge e 2 T 1, and a monomorphism G(e) ! G(v) for each pair (e, v) where v is an endpoint of e. F* *or any such tree of groups (G, T ), we let GT denote the amalgamated free product * *of the groups G(v) over the G(e), as described in [Se, xI.4.4]. Thus GT is the free pr* *oduct of the groups G(v) for all vertices v 2 T 0, modulo the relations given by the inc* *lusions of groups G(e) for e 2 T 1into the groups of the endpoints of e. Alternatively, one can regard T as a category whose set of objects is the di* *sjoint union of T 0and T 1, and with a pair of morphisms w e ! v for each edge e 2 T* * 1 with endpoints v, w. Then G is a functor from T to the category Gr+ of groups * *and monomorphisms, and GT = colim---!T(G). 16 CARLES BROTO, RAN LEVI, AND BOB OLIVER In this paper, we will be considering only finite trees of finite groups; i.e* *., pairs (G, T ) where T is a finite tree, and G(v) is a finite group for each v 2 T 0.* * Our goal in this section is to find some conditions on G and T which ensure that the gro* *up GT gives rise to a saturated fusion system and associated centric linking system. If (G, T ) is a tree of groups, and G = GT , then we let eTdenote the graph w* *ith vertex and edge sets fi a Te0 = (gG(v), v) fiv 2 T 0, g 2 G = G=G(v) x {v} v2T 0 fi a Te1 = (gG(e), e) fie 2 T 1, g 2 G = G=G(e) x {e} e2T 1 (with the obvious choices of endpoints). Equivalently, eT= hocolim-----!T(G=-).* * By [Se, Theorem I.9, p. 38], eTis a tree upon which G acts with orbit graph T , with fu* *ndamen- tal domain which can be identified with T (the subtree spanned by vertices (1G(* *v), v)), and with isotropy subgroups on the fundamental domain given by G. For any pair of groups H, G, let Rep (H, G) = Hom (H, G)= Inn(G), and let [ff* *] 2 Rep(H, G) be the class of ff 2 Hom (H, G). If (G, T ) is a tree of groups, and * *H is any finite group, we let Rep(H, G) be the graph with vertex and edge sets fi 0 Rep (H, G)0= (v, [ff]) fiv 2 T , [ff] 2 Rep(H, G(v)) fi 1 Rep (H, G)1= (e, [ff]) fie 2 T , [ff] 2 Rep(H, G(e)) . When ff 2 Hom (H, G(x)), where x is a vertex or edge in T , we write (x, [ff]) * *for the pair (x, [ff]), where [ff] is the class of ff in Rep(H, G(x)). Alternatively, i* *f we regard T as a category, then Rep (H, G) = hocolim-----!Rep(H, G(-)). T Lemma 4.1. Fix a finite tree of finite groups (G, T ). Set G = GT = colim---!(* *G), and let eT be as above. Then the following hold for any vertex v* of T and any su* *bgroup H G(v*): (a)The connected component of Rep(H, G) which contains (v*, [inclG(v*)H]) is is* *omorphic (as a graph) to eT H=CG(H). (b)The natural map ~= H :ss0 Rep (H, G) -----! Rep (H, G) is a bijection. In particular, for x a vertex or edge of T and ff 2 Hom (H,* * G(x)), (x, [ff]) lies in the connected component of the vertex (v*, [inclG(v*)H]) i* *f and only if ff 2 Hom G(H, G(x)). Proof.When x is a vertex or edge of T , we write Gx = G(x) for short: the isotr* *opy subgroup at x of the G-action, when we regard T as a subtree (a fundamental dom* *ain) of eT. If K G and gK 2 (G=K)H , then H gKg-1, and so we can regard c-1g:x 7! g-1xg as a homomorphism from H to K whose class in Rep (H, K) depends only on the coset gK. Hence it makes sense to define fH :eT H-----! Rep (H, G) A GEOMETRIC CONSTRUCTION OF SATURATED FUSION SYSTEMS 17 by sending each vertex (gGv, v) to the pair (v, [c-1g]) and each edge (gGe, e) * *to the pair (e, [c-1g]). We claim the following hold: (i)Im (fH ) is the connected component of (v*, [inclGv*H]) in Rep(H, G). (ii)A vertex (v, [ff]), for ff 2 Hom (H, Gv), lies in Im(fH ) if and only if th* *e composite ff H ___! Gv G is G-conjugate to the inclusion. (iii)fH induces an isomorphism of graphs (TeH)=CG(H) ~=Im(fH ). Point (ii) is immediate. If (e, [ff]) is an edge of Rep (H, G) with endpoint fH (gGv, v) = (v, [c-1g])* *, then v is an endpoint of e, and [ff] = [c-1g] 2 Rep (H, Gv). Thus ff = c-1hc-1g= c-1ghfo* *r some h 2 Gv, and (e, [ff]) = fH (ghGe, e). So an edge of Rep (H, G) lies in Im (fH * *) if one of its endpoints lies in Im(fH ), and thus Im(fH ) is a union of connected comp* *onents of Rep (H, G). Since eT His a tree by [Se, xI.6.1], Im(fH ) is nonempty and con* *nected, hence is a connected component of Rep(H, G), and this finishes the proof of (i). Two vertices (gGv, v) and (hGw, w) are sent to the same vertex of Rep(H, G) i* *f and only if v = w and [c-1g] = [c-1h] in Rep(H, Gv). This last condition is equival* *ent to saying that h 2 CG(H)gGv; i.e., that (gGv, v) and (hGw, w) are in the same CG(H)-orbit* *; and thus (iii) holds. Points (i) and (iii) together imply (a). By (ii), for any ff 2 Hom (H, G), H sends the connected component of a vert* *ex (v, [fi]) in Rep(H, G) to [ff] if and only if (v, [fiff-1]) 2 Im(fff(H)). Hence* * by (i), -1H([ff]) contains exactly one connected component. This shows that H is a bijection, * *and proves (b). We originally discovered the following theorem as a special case of Theorem 2* *.1 (and of Theorem 3.3), and it was certainly motivated by those results. However, sinc* *e it also has a more elementary proof which does not use certain deep theorems in homotopy theory, we give both proofs here. Theorem 4.2. Fix a prime p and a finite tree of finite groups (G, T ). Fix a ve* *rtex v* of T , set G* = G(v*) for short, and choose S 2 Sylp(G*). Set G = GT = colim--* *-!(G), F = FS(G), and L = LcS(G). Assume the following hold: (a)For each vertex v 6= v* of T , if e is the edge adjacent to v in the (unique* *) minimal path from v to v*, then [G(v) : G(e)] is prime to p. (b)If P S is F-centric (equivalently, if Z(P ) is a maximal p-subgroup of CG(* *P )), then the component of (v*, [inclG*P]) in the graph Rep (P, G) is a tree. (c)If some P S is G-conjugate to an essential p-subgroup of G(v) for any vert* *ex v, then P is F-centric. Then F is a saturated fusion system over S, and L is a centric linking system a* *ssociated to F. Proof.Let eT be as defined above: the tree upon which G acts with orbit space a* *nd fundamental domain T . When x is a vertex or edge of T , we write Gx = G(x) for short. 18 CARLES BROTO, RAN LEVI, AND BOB OLIVER We first show how the theorem follows as a special case of Theorem 3.3, appli* *ed to the action of G on eT. Condition 3.3(b) follows from condition (b) here, togeth* *er with Lemma 4.1(a); while condition 3.3(c) follows from condition (c) here. It remains to describe how condition 3.3(a) follows from condition (a) here. * * Let P G be any finite p-subgroup; we must show that eT Pcontains some vertex in t* *he orbit of (1G*, v*) in eT. Since eT is a tree, the fixed point set of the P -ac* *tion is also a tree, and hence its image in the orbit tree T is nonempty and connected. Let* * v be the vertex in that image which is closest to v*. If v 6= v*, then there is som* *e g 2 G such that gGv 2 (G=Gv)P, and hence g-1P g Gv. Let e be the edge adjacent to v* * on the minimal path from v to v*; then [Gv : Ge] is prime to p by (a), and hence t* *here is g0 2 G such that g0-1P g0 Ge. Then the edge (g0Ge, e) is in eT P, which contr* *adicts the original assumption about v. This shows that v = v*, and thus that some ver* *tex of the form (gG*, v*) is in eT P. Since this theorem also has a more elementary algebraic proof, we give that h* *ere. We first note that the argument just given also shows: (a0)For each vertex v in T and each p-subgroup P Gv, P is G-conjugate to a subgroup of S. By a proof identical to Step 2 in the proof of Theorem 3.3, we show (using (c* *)) that every morphism in F is a composite of restrictions of morphisms between F-centr* *ic subgroups. Hence by [BCGLO1 , Theorem 2.3], F is saturated if it satisfies ax* *ioms (I) and (II) in Definition 1.2 for all F-centric subgroups P S. So it remains to * *prove (I) and (II) for F-centric subgroups, and to prove that L is a centric linking * *system associated to F. For any H G*, let Rep (H, G)* be the connected component of (v*, [inclG*H])* * in Rep(H, G). By Lemma 4.1(b), if x is any vertex or edge in T , and ' 2 Hom (H, G* *x), then (x, [']) lies in Rep(H, G)* if and only if ' 2 Hom G(H, Gx). Proof of (I) for F-centric subgroups. Let P S be any subgroup which is F- centric and fully normalized in F. By Lemma 4.1, there is a bijection ss0(Rep (* *P, G)) ~= Rep(P, G) which sends Rep(P, G)* to [inclGP], and which is equivariant with res* *pect to the Aut(P )-action on both sets. Thus AutF (P ) = AutG (P ) is the isotropy sub* *group of Rep(P, G)* 2 ss0(Rep (P, G)) under the Aut(P )-action. In particular, AutF (* *P ) leaves Rep(P, G)* invariant. Since Rep (P, G)* is a tree by (b), and since every acti* *on of a finite group on a tree has a fixed point, there is a vertex (v, [ff]) in Rep * *(P, G)* which is fixed by Aut F(P ). Thus ff 2 Hom G (P, Gv). Set P 0= ff(P ) Gv, s* *o that AutGv(P 0) = ff AutF(P )ff-1. Fix Q02 Sylp(NGv(P 0)). By (a0), there is Q S w* *hich is G-conjugate to Q0. Fix fi 2 IsoG(Q0, Q), and set P 00= fi(P 0). Then |NS(P )| |NS(P 00)| |Q0| 0 | AutS(P )| = ________ ________ _______= | AutQ0(P )| ; |Z(P )| |Z(P 00)| |Z(P 0)| where the first inequality holds since P 00is G-conjugate (hence F-conjugate) t* *o P and P is fully centralized in F. Since AutQ0(P 0) 2 Sylp(Aut G(P 0)) and AutG (P 0) ~* *=AutG (P ), this proves that AutS(P ) 2 Sylp(Aut G(P )), and finishes the proof of (I). Proof of (II) for F-centric subgroups. Fix ' 2 Hom F (P, S) where P is F-centr* *ic, and set N' = {g 2 NS(P ) | 'cg'-1 2 AutS('(P ))} and K = AutN'(P ). A GEOMETRIC CONSTRUCTION OF SATURATED FUSION SYSTEMS 19 We claim that Res K Im Rep (N', G)* ----! Rep (P, G)* = Rep(P, G)* . (1) Clearly, if x is a vertex or edge in T , then the restriction of any fi 2 Rep(N* *', Gx) lies in Rep(P, Gx)K , so the problem is to prove that Rep(P, G)*K lies in the image.* * By (b), Rep(P, G) is a tree, and hence the fixed point set of the finite group K is als* *o a tree. So to prove (1), it suffices to show, for any edge (e, [ff]) in Rep(P, G)*K and* * any vertex (v, [fi]) in Rep (N', G)* such that (v, [fi|P]) is an endpoint of (e, [ff]), th* *at (e, [ff]) also lies in the image of the restriction map. In this situation, v is an endpoint of e, and we regard Ge as a subgroup of G* *v as usual. Then fi|P = cgOff for some g 2 Gv. Set P 0= ff(P ) and K0= ffKff-1 Aut* *(P 0) for short, and consider the subgroup Ne = {a 2 NGe(P 0) | ca 2 K0}. Fix Q 2 Sylp(Ne). Then AutNe(P 0) = K0 since (e, [ff]) is fixed by K, and AutQ * *(P 0) = K0 since K0 is a p-group. Also, since fi|P = cg Off, gQg-1 fi(N').CGv(fi(P ))* *. Since fi(P ) is p-centric in Gv, gQg-1 and fi(N') are both Sylow p-subgroups of this * *last group. _ -1 Hence there is h 2 CGv(fi(P )) such that hgQg-1h-1 = fi(N'). Set ff= chg Ofi 2 _ -1 * * _ Iso(N', Q). Then ff|P = cg Ofi|P = ff since h centralizes fi(P ), so Res((e, * *[ff])) = _ _ * * _ (e, [ff]). Also, (v, [ff]) = (v, [fi]) is an endpoint of (e, [ff]) since hg 2 G* *v, so [e, ff] is an edge in Rep(N', G), and this finishes the proof of (1). Now, (v*, [']) 2 Rep(P, G)* by Lemma 4.1(b), and is fixed by the K action by * *defi- nition of N'. So by (1), there is _ 2 Hom (N', G*) such that (v*, [_]) is in Re* *p(N', G)* and [_|P] = ['] in Rep (P, G*). Thus _ 2 Hom G(N', G*), and _|P = cg O' for some g 2 G*. By axiom (II) for the saturated fusion system FG*(S), c-1g= ' O(_|P)-1 _ def extends to some O 2 Hom G*(_(N'), S), and hence ' = O O_ 2 Hom G(N', S) extends '. This finishes the proof of (II) for F-centric subgroups. L is a centric linking system. If P is F-centric, then by point (b) and Lemma 4.1, CG(P )=Z(P ) acts on the tree eT Pwith orbit space a tree. Furthermore, s* *ince Z(P ) is maximal among finite p-subgroups of CG(P ), all isotropy subgroups of * *this action are finite of order prime to p. Hence by I.10, p.39]Serre, CG(P )=Z(P )* * is an amalgamated product of finite groups of order prime to p taken over a finite tr* *ee. Such a group is clearly p-perfect (it is generated by elements of order prime t* *o p); and Mayer-Vietoris sequences for the homology of amalgamated products (cf. [?, xVII* *.9]) show that Hi(CG(P )=Z(P ); Fp) = 0 for all i > 0. So by Lemma 1.5, L = LcS(G) i* *s a centric linking system associated to F. The most difficult hypothesis to check in the above theorem is (b). For this * *reason, we give here some equivalent formulations. The equivalence of the first three c* *onditions is implicit in the above proof, but we make them more explicit here. Lemma 4.3. Fix a finite tree of finite groups (G, T ), and set G = GT = colim--* *-!(G). Choose a vertex v* of T , and a subgroup H G(v*). Then the following three co* *nditions are equivalent: (1)The abelianization of CG(H) is finite. (2)CG(H) is a finite amalgamated product of finite groups. (3)The component of (v*, [inclG(v*)H]) in the graph Rep (H, G) is a tree. 20 CARLES BROTO, RAN LEVI, AND BOB OLIVER Furthermore, if (1)-(3) hold, then: (4)There is a vertex v 2 T 0and an element x 2 G, such that xHx-1 G(v) and Aut G(xHx-1) = AutG(v)(xHx-1). Proof.Let eTbe the tree upon which G acts with orbit space and fundamental doma* *in T , as in Lemma 4.1. Again, when x is a vertex or edge of T , we write Gx = G(x* *) for short. By Lemma 4.1, the component of Rep (H, G) which contains (v*, [inclGv*H]* *) can be identified with eT H=CG(H). If this orbit graph is a tree, then by [Se, Theo* *rem I.10, p.39], CG(H) is an amalgamated product of finite groups taken over a finite tre* *e; and in particular, its abelianization is finite. If the orbit graph eT H=CG(H) is n* *ot a tree, then by [Se, Corollary 1, p. 55], there is a surjection of CG(H) onto its funda* *mental group, an infinite free group, and hence the abelianization of CG(H) is not fin* *ite and CG(H) is not a finite amalgamated product of free groups. This proves the equiv* *alence of (1), (2), and (3). Now assume that (3) holds, and thus (by Lemma 4.1) that eT H=CG(H) is a tree.* * The finite group AutG (H) ~=NG(H)=CG(H) acts on this tree, and hence fixes some ver* *tex (cf. [Se, xI.6.1]). Assume the orbit of the vertex (a-1Gv, v) is fixed by Aut* * G(H); in particular, aHa-1 Gv since aGv 2 (G=Gv)H . Also, each ff 2 AutG (H) is of the* * form ff = cg for some g 2 NG(H) which fixes the vertex (a-1Gv, v) in eT H, which imp* *lies that aga-1 2 Gv. This shows that AutG (aHa-1) = AutGv(aHa-1), and thus that (4) holds. As shown in [AC ], the fusion systems FSol(q) constructed in [LO ] by the sec* *ond and third authors are the fusion systems of certain amalgamated products Spin7(q) ** *K, B where B is the normalizer in Spin7(q) of a certain elementary abelian 2-subgrou* *p of rank 2, and K contains B with index 3. The proof in [LO ] that these fusion sys* *tems are saturated is very long and technical, and so it is natural to wonder whethe* *r or not this could be shown as an application of Theorem 4.2. As seen in [LO ] or [AC ]* *, when F = FSol(q) and S 2 Sylp(Spin7(q)), then there is an elementary abelian 2-subgr* *oup E S of rank 4 such that AutF (E) = Aut(E) ~=GL4(2). Hence if the saturation of F could be proven using Theorem 4.2, then by Lemma 4.3(4), some vertex of the t* *ree defining the amalgamated product would be fixed by an extension of E by Aut (E), and this is not the case. More precisely, this shows that condition (b) in Theo* *rem 4.2 fails to hold for this amalgamated product. So Theorem 4.2 cannot be applied in* * this case. 5.Examples We now look at some applications of Theorem 4.2, to produce explicit exotic f* *usion systems. These examples will all be based on Proposition 5.1 below, which in tu* *rn is a special case of Threorem 4.2. For any fusion system F0 over a p-group S, any collection of subgroups Q1, . * *.,.Qm S, and outer automorphism groups i Out(Qi) containing OutF0(Qi), let denote the fusion system over S generated by F0 and restrictions of automorphis* *ms in the i to subgroups of Qi. In other words, F = is the fusion * *system A GEOMETRIC CONSTRUCTION OF SATURATED FUSION SYSTEMS 21 over S such that for all P, Q S, Hom F(P, Q) is the set of composites '1 '2 'k-1 'k P = P0 ---! P1 ---! P2 ---! . . .---!Pk-2 ---! Pk-1 ---! Pk = Q such that each j, either 'jlies in Hom F0(Pj-1, Pj), or for some 1 i m, Pj-* *1, Pj Qi and 'j is the restriction of some ffi2 Aut(Qi) such that [ffi] 2 i. The following proposition is also a generalization of [BLO2 , Proposition 9.* *1]. Proposition 5.1. Fix a finite group G, a Sylow p-subgroup S G, and subgroups Q1, . .,.Qm S such that no Qi is G-conjugate to a subgroup of Qj for i 6= j. * * For each i, set Ki = Out G(Qi), and fix subgroups i Out (Qi) which contain Ki. * *Set F = . Assume for each i that (1)p - [ i:Ki]; (2)Qi is p-centric in G, but no proper subgroup P Qi is F-centric or an essen* *tial p-subgroup of G; and (3)for all ff 2 irKi, Ki\ ffKiff-1 has order prime to p. Then F is a saturated fusion system over S, and has an associated centric linki* *ng system. Proof.For each i, set Hi= NG(Qi) and Ti= Op(CG(Qi)). Then Tihas order prime to p since Qiis p-centric in Hi; and thus Qi.CG(Qi) = QiTi~= Qix Tiand Ki~= Hi=QiT* *i. We first construct a finite group Gi Hi such that Hi has index prime to p in* * Gi, Qi C Gi, and Out Gi(Qi) = i. By (3), Ki\ ffKiff-1 has order prime to p for all ff 2 irKi; and hence the restriction homomorphism Hj( i; Z(Qi)) ~=Hj(Ki; Z(Qi)) is an isomorphism for all j > 0 by the description in of the image in terms of * *stable (or G-invariant) elements (cf. [AM , Theorem II.6.6] or [?, Theorem III.10.3])* *. When j = 3, the injectivity of the restriction map tells us that the obstruction to * *the existence of an extension 1 ---! Qi-----! G0i-----! i---! 1 vanishes [McL , Theorem IV.8.7], since its restriction to Ki ivanishes. When * *j = 2, the group H2( i; Z(Qi)) ~=H2(Ki; Z(Qi)) acts freely and transitively on the set* *s of all such extensions of Qi by i or by Ki [McL , Theorem IV.8.8], and thus G0ica* *n be chosen to contain the group Hi=Ti. Now let Tio Kiand Tio ibe the "regular" wreath products: the semidirect prod* *ucts Ti|Ki|oKiand Ti| i|o iwhere Kiand ipermute the factors Tifreely and transitive* *ly. There is an obvious embedding of Tio Ki into Tio i; and by [Hu , I.15.9], ther* *e is an embedding of Hi=Qi (as an extension of Ti by Ki) into Tio Ki. We can thus rega* *rd Hi=Qi as a subgroup of Tio i. So if we define Gi to be the pullback of the maps G0i-----! i----- Tio i, then Gi sits in an extension 1 ---! Qix Ti| i|-----!Gi-----! i---! 1; and we can identify Hi (regarded as a pullback of Hi=Ti and Hi=Qi over Ki) as a subgroup of Gi with index prime to p. Also, QiC Gi and i= OutGi(Qi). We will apply Theorem 4.2 to the tree which has m + 1 vertices v*, v1, . .,.v* *m and edges eiconnecting v* to vi, and to the functor G(v*) = G, G(vi) = Gi, and G(ei* *) = Hi. 22 CARLES BROTO, RAN LEVI, AND BOB OLIVER Let bGdenote the amalgamated product of this tree of groups. Then F = FS(Gb), a* *nd it remains only to check that conditions (a), (b), and (c) in Theorem 4.2 hold. Co* *ndition (a) holds by (1). We next check condition 4.2(c). By definition of F, for any subgroup P S wh* *ich p- centric in G, P is F-centric unless there is some P 0 S which is F-conjugate t* *o P but not G-conjugate, in which case P must be G-conjugate to a proper subgroup P 0 * *Qi for some i. For each i, any Sylow p-subgroup of Hi, or any essential p-subgroup* * (hence radical p-subgroup) of Gi, must contain Op(Hi) Qi; and the Qi are all p-centr* *ic in G (hence F-centric) by assumption. Any essential p-subgroup P of G is p-centric* * in G; and hence is F-centric since by (b) again, no essential p-subgroup of G is p* *roperly contained in any Qi. This finishes the proof of (c) in Theorem 4.2. It remains to check condition 4.2(b). Fix an F-centric subgroup P S; we mu* *st show that the component of (v*, [inclGP]) in Rep(P, G) is a tree. The edges i* *n Rep(P, G) adjacent to (v*, [inclGP]) are of the form (ei, [ff]), for ff 2 Hom G (P, Hi). * * For any such edge, its other vertex (vi, [ff]) is the endpoint of a second edge (ei, [fi]) o* *nly if fi = cgOff for some g 2 GirHi. In particular, ff(P ) and gff(P )g-1 are both contained in * *Hi, and thus ff(P ) Hi\ g-1Hig. By (3), (Hi\ g-1Hig)=Qi has order prime to p, and hen* *ce ff(P ) Qi. Since Qi is a minimal F-centric subgroup, this implies that ff(P )* * = Qi. Since no two of the Qiare G-conjugate, this can occur for at most one i. We thu* *s have two possibilities: o P is not G-conjugate to any Qi. In this case, every edge in is adjacent t* *o the vertex (v*, [inclGP]), and is a tree. o P is G-conjugate to Qj for some fixed j 2 {1, . .,.m}. In this case, let 0 be the subgraph of all vertices sitting over v* or vj, and all edges sitting* * over ej. Each vertex of not in 0 sits over vi for some i 6= j, and by the above re* *marks is connected to 0 by a unique edge. Thus 0 is a deformation retract of . * *Each edge in 0 has the form (ej, [ff]) for some ff 2 Iso(P, Qi). If two edges (e* *j, [ff]) and (ej, [ff0]) have the same vertex (v*, [ff]) = (v*, [ff0]), then ff0= cgO ff * *for some g 2 G, so g 2 NG(Qi) = Hi, and the edges (ej, [ff]) and (ej, [ff0]) are equal. Thus* * no vertex in 0 over v* can be attached to two edges, and this proves that 0 (and hen* *ce ) is a tree. This finishes the proof of condition 4.2(b), and hence of the proposition. Note that (3) implies (1) in the above proposition; condition (1) has been ke* *pt for emphasis. Condition (3) means that the subgroup Kiis strongly embedded in iat the prime p (see Definition 3.2). This puts fairly restrictive conditions on Ki and i, e* *specially when p = 2. By a theorem of Bender [Be ], if has a strongly embedded subgroup at p = 2, then either its Sylow 2-subgroups are cyclic or quaternion, or there * *is a normal series A C B C where A and =B have odd order, and B=A is isomorphic to P SL2(q), Sz(q), or P SU3(q) for q some power of 2. The severe restrictions * *which this places on the groups involved when applying Proposition 5.1 with p = 2 hel* *p to explain why it seems unlikely that we could construct an exotic fusion system a* *t the prime 2 using this proposition, although we are not yet able to completely excl* *ude that possibility. The following lemma is a refinement of [BLO2 , Lemma 9.2], and will be used * *to show that certain fusion systems are not fusion systems of finite groups. When F is * *a fusion A GEOMETRIC CONSTRUCTION OF SATURATED FUSION SYSTEMS 23 system over the p-group S, a subgroup P S is strongly closed if no element of* * P is F-conjugate to an element of Sr P . The subgroup P is normal in F if each morph* *ism in F extends to a morphism between subgroups containing P which sends P to itse* *lf. If P is normal in F, then it is strongly closed, but not conversely. As usual, a finite group G is almost simple if it contains a normal, nonabeli* *an simple subgroup L C G such that CG(L) = 1. In otherwords, G can be identified with a subgroup of Aut(L), and G=L with a subgroup of Out(L). Lemma 5.2. Let F be a fusion system over a nonabelian p-group S. Assume, for ea* *ch subgroup 1 6= P S which is strongly closed in F, that (a)P is centric in S (i.e., CS(P ) = Z(P )); (b)P is not normal in F; and (c)P does not factorize as a product of two or more subgroups which are permuted transitively by AutF (P ). Then if F is the fusion system of a finite group, it is the fusion system of a * *finite almost simple group. Proof.Assume that F = FS(G) for some finite group G with S 2 Sylp(G), and that G is a subgroup of minimal order with this property. Let 1 6= L C G be a minim* *al nontrivial normal subgroup. Set P = L \ S 2 Sylp(L); then P is strongly closed* * in F. If P = 1 (i.e., L has order prime to p), then F is also the fusion system of* * G=L, which contradicts the minimality assumption. If L = P is an abelian p-group, t* *hen it is normal in F, which contradicts (b). Thus, since L is minimal, it is a pro* *duct of nonabelian simple groups isomorphic to each other (cf. [Go , Theorem 2.1.5]); a* *nd these must be permuted transitively by NG(L) = G since otherwise L is not minimal. Th* *en L must be simple by (c). Also, CG(L) \ S CS(P ) P by (a). Since CG(L) C G, this means it must have order prime to p (otherwise it would intersect every Sy* *low p-subgroup nontrivially); and this implies CG(L) = 1 by the minimality assumpti* *on (again since CG(L) C G). Thus G is almost simple; i.e., L C G Aut(L). We next focus attention on cases where Proposition 5.1 can be applied with Qi* *~= C2p (for p an odd prime). By [Hu , Satz III.14.23], a p-group S contains a centric * *subgroup of order p2 if and only if it has maximal class; i.e., if and only if it has ni* *lpotence class n-1 when |G| = pn. For odd p, the structure of p-groups of maximal class is des* *cribed in detail in [Hu , xIII.14], and include the following examples when p = 3. Example 5.3. Set p = 3, and let S be one of the following groups of order 34: fi 9 3 3 -1 -1 -3ff S0 = a, b, x fia = b = x = [a, b] = 1, xax = ab, xbx = ba . fi 9 3 3 -1 -1 3ff S00= a, b, x fia = b = x = [a, b] = 1, xax = ab, xbx = ba . Let !, j 2 Aut(S) be the automorphisms ( ba3 if S = S0 -1 !(a) = a-1, !(b) = x-1bx = !(x) = x ba-3 if S = S00, j(a) = a-1, j(b) = b-1, j(x) = x. For i = 0, 1, 2, set Ri= and Qi= . 24 CARLES BROTO, RAN LEVI, AND BOB OLIVER Then Ri ~=C23if S = S0 or if i = 0, and Ri ~=C9 if S = S00and i = 1, 2. Also, Qi is extraspecial of order 27, and has exponent 3 if S = S0 or i = 0 and expon* *ent 9 otherwise. All of these subgroups are invariant under !, while j leaves R0 an* *d Q0 invariant and switches R1 and R2. When S = S00, then the following fusion systems over S ); SL(R0)> and ); GL(R0)> are both saturated, and not fusion systems of any finite group. When S = S0, th* *en the following table describes different fusion systems over S via the automorphism * *groups OutF (P ) for P = S, Ri, or Qi, and where an asterisk marks those which are not* * fusion systems of any finite group: ____________________________________________________________________________ | Out F(S) |AutF(R0) |OutF(Q0) |AutF(R1) |AutF (R2) | group | |__________|_________|_________|_________|__________|_______________________| | | SL2(3) | _ | SL2(3) | SL2(3) | L (q) (v3(q 1) = 2) | |_________|_________|________|___________|_________|___3____________________ | | | _ | _ | SL2(3) | SL2(3) | * | |_________|________|_________|___________|_________|________________________| | | SL2(3) | _ | _ | _ | * | |_________|_________|________|__________|_________|_________________________| | |GL2(3) | _ | SL2(3) |L (q)o C2 (v3(q 1) = 2) | |__________|________|________|______________________|_3_____________________ | | | _ | GL2(3) | SL2(3) | 3D4(q) (v3(q2 - 1) = 1) | |__________|_______|__________|_____________________|_______________________ | | | _ | _ | SL2(3) | * | |__________|_______|_________|______________________|_______________________| | |GL2(3) | _ | _ | * | |__________|________|________|_____________________|________________________| Here, L+n(q) = P SLn(q) and L-n(q) = P SUn(q). Also, for any prime power q, 3D4* *(q) is the fixed subgroup of a certain "triality" graph automorphism of order 3 on Spi* *n8(q3). Proof.That these fusion systems are all saturated is a special case of Proposit* *ion 5.1, applied with G = So or G = So as appropriate. Let F be any of these fusion systems, and assume P C S is a proper strongly c* *losed subgroup. Then P (any normal subgroup contains the center). If F contains SL(Ri) for some i, then P Ri since is F-conjugate to the other subgroups* * of order 3 in Ri; and hence P Qi(the normal closure of Riin S). By similar reaso* *ning, if F contains SL(Qi) for some i, then either P = , or P Qi. Thus in all c* *ases listed above, the only nontrivial subgroups strongly closed in F are S, and pos* *sibly one of the Qi. Hence by Lemma 5.2, if F is the fusion system of a finite group,* * then it is the fusion system of a finite almost simple group G, which contains a normal* * simple group LfCiG withfSylowi3-subgroup S or Qi. If L contains a Sylow 3-subgroup Qi, then 3fi|G=L|fi| Out(L)|, and this is impossible by Lemma 5.4 below. It remains to consider the case where [G : L] is prime to 3, and thus where S* * 2 Syl3(L). By [GLS , Tables 5.3 & 5.6.1], none of the sporadic simple groups has * *Sylow 3-subgroup of order 34 and rank 2. If v3(|An|) = 4, then n = 9, 10, 11, and rk3* *(An) = 3. By [GLS , Table 2.2], the only simple groups of Lie type in characteristic 3 wh* *ose Sylow 3-subgroups have order 34 are the groups B2(3), and these also have 3-rank equa* *l to 3. Finally, using [GLS , Table 2.2] and [GL , 10-1 & 10-2], one checks that the on* *ly simple groups of Lie type whose Sylow 3-subgroups have order 34 and rank 2 are the gro* *ups L3(q) when v3(q - 1) = 2, U3(q) when v3(q + 1) = 2, and 3D4(q) when v3(q2 - 1) * *= 1. The precise fusion systems of these groups (and the fact that their Sylow subgr* *oups are isomorphic to S0) is determined directly, or with the help of the lists of * *maximal subgroups in [GLS , Theorem 6.5.3] (for L3(q)) and [Kl] (for 3D4(q)). For examp* *le, by A GEOMETRIC CONSTRUCTION OF SATURATED FUSION SYSTEMS 25 [Kl], there are subgroups 32.SL2(3) 3D4(q) and 31+2+.GL2(3) SL3(q).S3 3D4(q) (when q 1 (mod 3)), and this determines the structure of the fusion system of 3D4(q). It remains to prove the following lemma, which will also be used later. Lemma 5.4. There is no pair (L, p), where L is a finite simple group,fpiis an o* *dd prime, the Sylow p-subgroups of L are extraspecial of order p3, and pfi| Out(L)* *|. Proof.If L is a sporadic or alternating group, then | Out(L)| is a power of 2, * *so this is impossible. Thus L is of Lie type, and hence by [Ca , Theorem 12.5.1], Out * *(L) is generated by field, graph, and diagonal automorphisms. We refer to [Ca , 9.4.10* *, 10.2.4- 5, 14.3.2] for the orders of the simple groups of Lie type. The only simple gro* *ups with graph automorphisms of odd order are the groups D4(q) (with graph automorphisms of order 3), and |D4(q)| = q12(q8 + q4 + 1)(q6 - 1)(q2 - 1) is a multiple of 34* * for all q. If L has a field automorphism of order p, where p is an odd prime, then L is de* *fined over a field of order qp for some prime power q; if qpn 1 is divisible by p t* *hen it is divisible by p2, and the list of orders of groups of Lie type makes it clear th* *at this case is impossible. So if there is a pair (L, p) as above, then L must have a d* *iagonal automorphism of order p. The only simple groups of Lie type with diagonal automorphisms of order p 3* * are P SLn(q) (for p|(n, q - 1)), P SUn(q) (for p|(n, q + 1)), E6(q) (for p = 3|q - * *1), and 2E6(q) (for p = 3|q + 1). Of these, the only cases where the simple group has p* *-rank 2 occur when p = 3, and L = P SL3(q) (where 3|(q -1) and |L| = 1_3q3(q2-1)(q3* *-1)) or P SU3(q) (where 3|(q + 1) and |L| = 1_3q3(q2 - 1)(q3 + 1)). In both of thes* *e cases, v3(|L|) = 2v3(q 1) 6= 3. For the rest of the section, we let p be any odd prime, and consider the group fi p p p p S = a, b, c, x fia = b = c = x = [a, b] = [a, c] = [b, c] = 1, ff xax-1 = a, xbx-1 = ab, xcx-1 = bc . Set A = , Q = , and R = . Set x ffi -2 fi x = GL2(p) x Fp (uI, u ) fiu 2 Fp , and let [B, u] denote the class of the pair (B, u) for B 2 GL2(p) and u 2 Fxp. * *Define an action O: ___! Aut(A) as follows. Identify A with the additive group S2(p* *) of symmetric 2 x 2 matrices by setting a = 2000, b = 1110, and c = 0001; and let [B, u] 2 act by sending M to u.BMBt. These identifications are chosen so that* * the action of X def= 1011, 1 on A is precisely the action of x 2 S by conjugation.* * We __def can thus identify S as a Sylow p-subgroup of G = Ao . Then is the normaliz* *er in Aut (A) ~= GL3(p) of the orthogonal group GO3(p), for an appropriate choice * *of quadratic form on A. For a given pair of fusion systems F0 F over0the same p-group S, we say that F0 has index prime to p in F if AutF0(P ) Op (Aut F(P )) for all P S [BCGLO* *2 , Definition 3.1]. In [BCGLO2 , x5], we prove that for any saturated fusion sy* *stem F, 0 there is a unique minimal fusion subsystem Op (F) F of index prime to p. This terminology provides a convenient framework for describing the next result. 26 CARLES BROTO, RAN LEVI, AND BOB OLIVER Example 5.5. Fix an odd prime p, and let S be the group of order p4 defined abo* *ve, with subgroups A, Q, R S. Then the following hold. (a)There are unique saturated fusion systems FQ and FR over S such that OutFQ(S) ~=Cp-1x Cp-1, AutFQ(A) = , OutFQ (Q)= Out(Q) ~=GL2(p) OutFR(S) ~=Cp-1x Cp-1, AutFR(A) = , Aut FR(R)= Aut(R) ~=GL2(p) ; and Q is not FR-radical. 0 (b)Op (FQ) has index 2 in FQ, Out Op0(FQ)(Q) is the unique subgroup of index 2 * *in Out (Q) ~=GL2(p), and fi AutOp0(FQ)(A) = [B, 1] fiB 2 GL2(p) . For all p, FQ is the fusion system of Aut(P Sp4(p)) = P Sp4(p)o C2 (the exte* *nsion 0 by diagonal automorphisms), and Op (FQ) is the fusion system of P Sp4(p). 0 (c)Op (FR) has index (4, p- 1) in FR, Aut Op0(FR)(R) is the unique subgroup of * *index (4, p- 1) in Aut(Q) ~=GL2(p), and fi -1 x4 AutOp0(FR)(A) = [B, u] 2 fidet(B).u 2 Fp . 0 When p = 3, FR is the fusion system of 9 and O3 (FR) is the fusion system o* *f A9. When p = 5, FR is the fusion system of P L5(16) ~=P SL5(16)o C4 (the extens* *ion 0 by field automorphisms), and O5 (FR) is the fusion system of P SL5(16). When p 7, no fusion subsystem of index prime to p in FR is the fusion system of* * a finite group. Proof.Set G = Ao , and identify S with Ao G. Since N () is gener- ated by X = 1101, 1 together with elements u00v, w for u, v, w 2 Fxp, and* * since u00u, u-2 = 1, OutFQ (S) = OutFR (S) = OutG (S) = {[ju!v] | u, v 2 Fxp} ~=Cp-1x Cp-1, where [ju] and [!v] are the classes modulo Inn(S) of the automorphisms u -(u-1)=2 u2 1=u ju = c 100u, 1 : a7! a b7! b a c7! c x 7! x v v v !v = c I, v : a7! a b7! b c7! c x 7! x for all u 2 Fxp. Here, c(g) denotes conjugation by g (a 7! gag-1). Note also th* *e relation 2 2 -2 c u0u0-1, 1 = c 10u0-2, u = !u2j1=u2= !uju . It follows that OutG (Q) = = NOut(Q)(Out G(Q)). So we can apply Proposition 5.1 with m = 1 and Q1 = Q (and with G as above), to prove that the fusion system FQ is saturated. Similarly, AutG (R) = = NAut(R)(Aut G(R)), and so FR is saturated by Proposition 5.1 again. 0 p0 We next calculate Op (FQ). Let O* (FQ) FQ be the fusion subsystem gener- 0 ated by the automorphism groups Op (Aut F(P )) for P S. Consider the subgroup Out0FQ(S) C OutFQ (S) as defined in [BCGLO2 , x5.1]: fi * * ff Out0FQ(S) = ff 2 OutFQ (S) fiff|P 2 Mor Op0*(FQ)(P, S), some FQ-centricP * *S . A GEOMETRIC CONSTRUCTION OF SATURATED FUSION SYSTEMS 27 0 For u, v 2 Fxp, (ju!v)|Q 2 Op (Aut FQ(Q)) ~= SL2(p) if and only if v = 1; while 0 x2 0 (ju!v)|A 2 Op (Aut FQ(A)) = 0 if and only if v = 1=u 2 Fp . Thus Out FQ(S) = 0 {ju!v2}. This shows that Op (FQ) has index 2 in FQ, and has the form described * *in (b). 0 A similar argument shows that Op (FR) has index (4, p- 1) in FR, and has the * *form described in (c). 0 0 Thus for all F-centric subgroups P 0 S, AutF (P 0) contains Op (Aut FP(P )) * *(P = Q or R). Hence F has index prime to p in FP, and by [BCGLO2 , Theorem 5.4], 0 Op (FP) F FP. It is straightforward to check that the finite groups listed in (b) and (c) h* *ave the automorphism groups as indicated, and we have seen that this determines their f* *usion system. So it remains to show that the fusion systems in (c) are not fusion sys* *tems of finite groups for p 7. Let F be any of these fusion systems. If 1 6= P C S is strongly closed in F, * *then it must contain Z(S) = (any nontrivial normal subgroup intersects nontrivially* * with Z(S)); hence contains A (since the subgroups Aut F(A)-conjugate to generate* * A in all cases); and hence is equal to S since either Q or R is F-radical. So by * *[BLO2 , Lemma 9.2], if F is the fusion system of a finite group, it must be the fusion * *system of a finite almost simple group. More precisely, F = FS(G) for some G with norm* *al simple subgroup L C G of index prime to p such that CG(L) = 1. By a direct check through the list of finite simple groups, one sees that the following are the o* *nly simple groups which have Sylow p-subgroup isomorphic to S: o (any p) P Sp4(p) o (p = 3) P SL4(q) (q 4, 7 (mod 9)), P SU4(q) (q 2, 5 (mod 9)), P Sp6(q) (q 2, 4 (mod 9)), 7(q) (q 2, 4 (mod 9)), An (n = 9, 10, 11). o (p = 5) P SL5(q) (q 6, 11, 16, 21 (mod 5)), P SU5(q) (q 4, 9, 14, 19 (mo* *d 5)), Co1. By elimination, none of the FR,ifor p 7 is the fusion system of a finite grou* *p. In fact, in the above situation, if F is any saturated fusion system over S s* *uch that A is F-radical but not normal in F, then F is isomorphic to a fusion system F0 * *over S which has index prime to p in one of the fusion systems FQ or FR. To see this* *, set = AutF (A) GL3(p) for short, and let 0be the image of \ SL3(p) in P SL3(* *p). If 0has a nontrivial normal subgroup of order prime to p, then either the acti* *on on A is decomposable (which is impossible since the action of AutS(A) is indecomposa* *ble), or A splits as a sum of three subspaces which are permuted by . The latter ca* *se implies that Cp-1o 3, and thus is possible only if p = 3 and C C2 o 3 =* * . Otherwise, if 0 has no nontrivial normal subgroups of prime power order, then * *by [Bl, Theorem 1.1], 0 must be isomorphic to P SL2(p) or P GL2(p); and conjugate to the indecomposable representation of these groups described in [Bl, Lemma 6.* *3]. Hence up to conjugacy, \ SL3(p) contains 0 as a normal subgroup, and hence t* *hat NGL3(p)(P SL2(p)) = . In particular, every proper subgroup P S not contained in A is either F-con* *jugate to Q or R; or else p = 3 and P is cyclic or extraspecial of exponent 9 (in whic* *h case P cannot be F-radical). Hence since A is not normal in F, one of the subgroups * *Q or R must be F-radical. If P = Q or R is F-radical, then OutF (P ) Out(P ) ~=GL2* *(p) 28 CARLES BROTO, RAN LEVI, AND BOB OLIVER contains at least two subgroups of order p; any two such subgroups generate SL2* *(p); and thus OutF (P ) SL2(p). When p 5, there are also saturated fusion systems over S where A is not rad* *ical, but is not normal either. Fix any subset I {0, . .,.p - 1} with |I| 2, and * *choose Pi= or for each i 2 I. Let FI be the fusion system over S * *generated by OutFI(S) = ~=C2p-1(where ju and !u are defined as above), and OutFI(Pi* *) = Out(Pi) for all i 2 I. (This depends not just on I but also on the choice of th* *e Pi.) These are saturated fusion systems by Proposition 5.1, and have no proper stron* *gly closed subgroups. So by the list of simple groups with Sylow subgroup S given a* *bove, these systems are all exotic. We look more closely only at the case where |I| = 1 and Pi~= R. In this case,* * there is a proper strongly closed subgroup. Example 5.6. Fix an odd prime p, and let S be the group of order p4 defined abo* *ve. For u = 1, . .,.p - 1, let !u, ju 2 Aut(S) be the automorphisms !u(a) = au, !u(b) = bu, !u(c) = cu, !u(x) = x; 2 1=u ju(a) = a, ju(b) = bua-(u-1)=2, ju(c) = cu , ju(x) = x . Set = {!ujv| u, v = 1, . .,.p - 1} and 0 = {!uju | u = 1, . .,.p - 1* *}. Set R = ~= C2p; a subgroup invariant under each !i and ji. Then the fus* *ion systems F = and F0 = are both saturated, and neither is the fusion system of a finite group. Proof.The fusion systems F and F0 are saturated by Proposition 5.1, applied with G = So or So 0, respectively. If P 6= 1 is strongly closed in F or F0, then P = Z(S) (since any nontr* *ivial normal subgroup intersects nontrivially with the center), hence P R, and hen* *ce P Q = since bx is S-conjugate to x. Thus P = Q or S. Hence by Lemma 5.2, if F or F0 is the fusion system of a finite group G, then we can assume th* *at there is a normal simple subgroup L C G such that CG(L)f=i1f(soiG=L Out(L)), and su* *ch that L S or L \ S = Q. If L \ S = Q, then pfi|G=L|fi| Out(L)|, and this is im* *possible by Lemma 5.4. Thus S 2 Sylp(L). In the proof of Example 5.5, we listed all simple groups L with Sylow p-subgr* *oup isomorphic to S for some odd p. In all cases, the (unique) abelian subgroup of * *index p in S is radical in L, and thus FS(L) is not contained in F. References [AM] A. Adem & J. Milgram, Cohomology of finite groups, Springer-Verlag (199* *4) [As] M. Aschbacher, Finite group theory, Cambridge Univ. Press (1986) [AC] M. Aschbacher & A. Chermak, A group-theoretic approach to a family of 2* *-local finite groups constructed by Levi and Oliver (in preparation) [Be] H. Bender, Transitive Gruppen gerader Ordnung, in denen jede Involution* * genau einen Punkt festl"asst, J. 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Stancu, Equivalent definitions of fusion systems, preprint Departament de Matem`atiques, Universitat Aut`onoma de Barcelona, E-08193 Bel- laterra, Spain E-mail address: broto@mat.uab.es Department of Mathematical Sciences, University of Aberdeen, Meston Building 339, Aberdeen AB24 3UE, U.K. E-mail address: ran@maths.abdn.ac.uk LAGA, Institut Galil'ee, Av. J-B Cl'ement, 93430 Villetaneuse, France E-mail address: bob@math.univ-paris13.fr