1 ON THE K-THEORY OF THE CLASSIFYING SPACE OF A DISCRETE GROUP Alejandro Adem* Mathematics Department University of Wisconsin Madison,WI 53706 Dedicated to the memory of Jose Adem (1921-1991). x0. INTRODUCTION Let G be a finite group. By considering the action of G on itself by conjug* *ation, one arrives at the class equation for G, ! X ! 1 1 !1jGj= !jC(g)j (0:1) g(g)6=1 where g ranges over conjugacy classes of elements in G, with centralizer C(g). Now assume that is a discrete group of finite virtual cohomological dimens* *ion with centralizers of elements of finite order in homologically finite (for exa* *mple SLn(Z)).!Brown![B2] proved the generalization of (0.1) to : ! ! X !! "\() \()= \(C(fl)) (0:2) ! f(fl)l6=1 ! ! where!fl!ranges over conjugacy classes of elements of finite order in , "\deno* *tes the!topological!Euler characteristic and\ the usual group-theoretic version (if* * 0is ! !! 0) a!torsion-free!subgroup of finite indexin then by definition \() = "\([:0]). T* *he ! 2 point of (0.2) is that the difference between \"() and \()is determined by the torsion in , in particular if is torsion-free they coincide. Now let us recall a well-known result in algebra: let R(G) denote the chara* *cter ring of a finite group G. Then No. of conjugacy dimC R(G) C 1= classes of non-trivial (0:3) elements in G : The term on the left can be identified with dim CKG () C,the dimension of the equivariant K-theory of ap oint. In this paper we will use this interpretation* * to outline an approach for generalizingthis formula to infinite groups of finite * *virtual cohomological dimension (with suitable finiteness conditions), motivated by (0.* *2). By a result due to Serre [S], will act on a finite dimensional, contractib* *le space X, with finite isotropy, andsuch that XH ' for all H finite. Fix an extension 1 ! 0! !ssG ! 1 with 0 torsion-free, G finite. Then 0 acts freely on X, G acts on X=0such that the finite subgroups in correspond under ssto the isotropy subgroups in G. We have Theorem 3.1 M KG(X=0) C = K (B0)G C K (B(C (fl) " 0))Hfl C fl2Cf1g where C is the set of conjugacy classes of elements of finite order in , and Hf* *l= C(fl)=C(fl) " 0, a finite group. * * == An immediate corollary of this is the formula X \(KG(X=0) C) "\() = "\(C(fl)) : (0:4) fl2C f1g 3 The ingredient which makes (0.3) useful is the fact thatR(G)^ = K (BG) (the completion theorem). Let act on X EG diagonally and through !ssG; then B ' X=0GEG, whence the completion theorem implies KG (X=0)^ = K (B), and so, up to IG-adic completion, the preceding results provide information on K (B). For example, if G is a p-group completion is p-adic completion, and hence (calculating over the p-adics Cp), X \(K (B) Cp) "\() = \"(C(fl)): (0:5) fl2Cf1g We point out that will always contain subgroups of this form (G a p-group) with finite index. The key technical device which we use is a result of N. Kuhn [K] expressing KG(Y ) C as a sum of K (Y =C(g)) C,as g ranges over conjugacy classes of elements in G. The paper is organized as follows: in x1 we describe the -complex X; in x2 * *we outline the main properties of equivariant K-theory, in x3 we prove our result * *and in x4 we provide examples, as well as a p-local version of the results in x3: * *using p-adic K-theory we obtain an exact formula (4.2). Formula (3.1) indicates that the K-theory of B can be calculatedgiven enough information about its elements of finiteorder and their centralizers. It is int* *eresting to compare this with results obtained by Brown (see [B3]) for computing the (hi* *gh- dimensional) cohomology of B. In the general situation there is a spectral sequ* *ence involving the cohomology of thenormalizers of the finite subgroups, convergingto the cohomology of in sufficiently high dimensions (or in any dimension if Farr* *ell Cohomology is used). This can be difficult to deal with, except in the rank one situation, where the spectral sequence only has one line. In contrast, K (B) C seems to be much more accessible interms of subgroup data, a fact which is of c* *ourse 4 spectral sequence or other techniques,this can (in some cases) yield informatio* *n on the cohomology. x1. GROUPS OF FINITE VCD Definition 1.1: A discrete group is said to have finite virtual cohomological dimension (vcd < 1) if there is a subgroup of 0 of finite index such that 0 has finite cohomological dimension. * *== Examples of this type of group include arithmetic groups (such asS Ln(Z)) a* *nd mapping class groups. The key geometric ingredient in the analysis of these groupsis a result due* * to Serre [S]: Theorem 1.2(Serre) If is a group with vcd < 1, then there exists a finite dimensional, proper contractible -complex Xwith the following additional property: XH is contractib* *le for all finite subgroups H . == This can be easily summarized as follows. Fix 0 a normal subgroup of finite cohomological dimension. Then by the well-known result of Eilenb erg and Ganea [EG] we can find a finite dimensional K (0; 1) complex, whose universal cover X0is a contractible 0-complex. If r = [ : 0], then acts faithfully as a group of automorphisms of the principal0-bundle ! =0and hence embeds in the full automorphism group rT (0)r. As the latter group acts on (X0)r, we obtain a -action on X =(X 0)r. One checks that this satisfies all the condition* *s. In particular [B] if H is finite, XH = (X0)k, wherek = [ : 0]=jH j, hence it* * is contractible. In a more abstract language, X can be regarded as an "E (0)-space." We recall what that means. Given H /G, let FG (H) be the family of all subgroups 5 (1) X K = ; for all K62 FG (H) (2) X K is contractible for all K2 FG(H ). The orbit space EG (H)=H is called a BG H space,and as a G=H -space is unique up to weak G=H-equivalence. In our situation F (0) consists of all finite subgroups in , whence Serre's construction is a particular finite dimensional model for the generalized class* *ifying space B 0; (see [tD] for more on this). Let us now fix an extension 1 ! 0! ! G ! 1 where cd0< 1, jGj < 1. We willnow make a detailed examination of the fixed- point sets Xhfli, where fl 2 is an element of finite order, hfli the subgroup * *it generates, and X a complex as in 1.2. First we have Lemma 1.3: Let g2 G, then a a (X=0)hgi= XH =0" N(H) ' B(0" N(H)) H2C H2C where C ranges over all 0-conjugacy classes of finite subgroups H mapping onto hgi. Proof: This is explicitly described in [B], pg. 267, and follows from looking * *at in- verse images under ss: ! G and the additional fact that XH is a free, contract* *ible 0" N(H)-space. == Now consider the C(g)-action on this fixed-point set. We describe its orbit space as follows Lemma 1.4: a a (X=0)=C (g) = X=C(fl ) 6 where the elements range over conjugacy classes of elements of finite order in * *G and respectively. ` Proof: Let Cg(g)= ss1 (C(g)), then this group acts on fl2C"Xfl, where "Cis t* *he collection of all elements of finiteorder in mapping onto g. If x 2 Xfl, c 2 g* *C(g), take cx; clearly cx 2 Xcflc1, and ss (cfl c1 ) = g. We now take the orbit space 0 1 a a @ X flA ffigC(g)= X fl=gC(g)"C(fl) fl2"C fl2C where now Cg(g)-conjugacy classes of C = fl 2 offinite order, ss(fl) =:g Clearly C(fl) gC(g), hence gC(g)"C(fl) = C(fl ). Next we claim that if fl 2 of finite order,ss(fl) = g ,and if fl0 = fl1flf* *l11 also satisfying ss(fl0) = g, then fl1 2Cg(g). For we have ss(fl1)gss(fl1)1 = g; hence ss(fl1) 2 C(g) and so fl12 Cg(g). As elements of finite order in are mapped injectively under ss,as we range over conjugacy classes of elements of G, we obtain the asserted disjoint union.* * == We have the following cohomological corollary of 1.4. Corollary 1.5: 0 1 a M H @ (X=0)hgi=C (g);QA = H (BC (fl ); Q) : (g) (fl) finite order Proof: We have extensions 7 where Hflis finite. Hence we obtain H(B C(fl);Q) = H (B0" C(fl);Q)Hfl= H (X=C(fl);Q); the last equality because Xis a contractible C(fl)-space with a free C(fl) * *" 0- action. * *== x2. EQUIVARIANTK-THEORY We recall some well-known facts about G-equivariant K-theory, for G a finite group. Our main reference is [A-S1]. In this section we assume Y isa compact G-space. KG (Y) is defined by using G-vector bundles overY . Using the natural map Y ! , we have a homomorphism KG () ! KG (Y ) with which KG(Y ) is a (we will assume finitely generated) KG ()-module. Recall that KG() = R(G), the complex character ring of the finite group G. Denote by YG = (Y E G)=G the usual Borel construction. If F is a G-vector bundle on Y, then (F EG)=G is a vector bundle on YG; the assignment F 7! (F EG)=G is additive hence it induces a homomorphism ff : KG(Y ) ! K (Y G EG): Here the term on the right is defined as limKG (Y EG(n)) for a suitable filtration of EG,and the map above can be alternatively construc* *ted as follows: the natural projection Y EGn ! Y induces 8 In particular, if IGn denotes the kernel of R(G) = KG () ! KG (EGn ),then ffn factors KG (Y )=IGnKG (Y ) ffn!K (Y G EG): The main result due to Atiyah and Segal concerning this map is Theorem 2.1 (Completion Theorem) Let Y be a compact G-space such that KG (Y) is finite over R(G). Then the homomorphisms ffn : KG(Y )=IGn ! KG(Y EGn) induce an isomorphism of pro-rings. == In particular if IG ae R(G) is the augmentation ideal, and KG(Y) is endowed with the IG-adic topology, then 2.1 can be rephrased as saying that ff : KG(Y ) !K (Y G EG) induces an isomorphism of the IG-adic completion of KG(Y) with K (Y G E G). The principle then is that KG(Y ) can be approximated by K (Y GE G). To conclude this section we describe a more recentresult due to N. Kuhn [K] relating KG(Y ) to the K-theory of Y =C (g). Namely, he proves that there is* * an isomorphism M KG(Y ) C = K (Y =C(g)) C (2:2) (g) where the sum is taken over all conjugacy classes of g 2 G (note the case Y = , we recover the formula for dim CR(G) C). This correspondence can be outlined fifi as follows. Let F be a G-vector bundle on Y; then on F Y the element g still fifi acts, leaving points in the base fixed. Therefore F Y will split as a direct* * sum of vector bundles corresponding to the eigenspaces of g; putting the eigenvalue in* * the 9 take invariantsto obtain an element in K(Y )C (g)C = K(Y =C(g)) C; the same holds for K1G(Y ). This description is also given in [H-H] , [A-S2],wh* *ere it is pointed out that this is related to work in string theory concerning orbifol* *d Euler characteristics. x3. APPLICATION TO K (B) The goal of this section will be to use the preceding results to obtain ana* *pprox- imation to K (B), for a discrete group of finite v.c.d., withsuitable finitene* *ss assumptions. As before, fix an extension 1 ! 0! ! G ! 1 where cd 0< 1, jGj < 1. Let X be an admissible -complexas b efore, with suitable (e.g. compactness) finiteness assumptions on X=0. Then, using ! G, acts diagonally on X EG without any non-trivial isotropy. As X EG , this means B ' (X EG)= = X=0G EG and hence K (B) = K (X=0G EG): It is now evident that the completion theorem provides amethod for approach- ing K (B), namely via the map KG(X=0) ff!K (B) which will induce an isomorphism KG(X=0)^= K (B). 10 Theorem 3.1: M KG (X=0) C = K (B(C(fl) " 0))Hfl C (fl) where fl ranges over conjugacyclasses of elements of finite order in , and Hfl= C(fl)=C(fl) " 0, a finite group. Proof: Simply combine (2.2) with (1.4) and identify K (X=C(fl)) C =K ([X=C(fl) " 0]=Hfl) C =K (B(C(fl) " 0))Hfl C : == From this we derive (if each C(fl) is homologically finite): Corollary 3.2. X \(KG (X=0)) = "\(C (fl)): (fl) finite order Proof: Use that fact that the Euler characteristic of K (Y ) equals \(Y ). * * == What we have is an expression for KG(X=0) involving only centralizers of elements of finite order in ,which after IG-adic completion determines K (B). In some cases this completion process can be straightforward, in particular if * *G is a p-group, it is just p-adic completion.Computing ranks over Cp leads to Theorem 3.3: Let be a discrete group of finite v.c.d. such that (a) the cen- tralizers C(fl) are of finite homological type for all fl 2 of finite order and* * (b) there is a torsion-free subgroup 0, normal in and such that =0=G is a finite p-group. Then X \(K (B) Cp) "\() = "\(C(fl)) : (fl) fl offinite 11 == Remark: An elementary case ofthe above results occurs when is torsion-free; then X= ' B and hence \(K (B)) = "\(); on the right-hand sideit vanishes because there is no torsion. Recall now the usual Euler characteristic for discrete groups: \() ="\(0)=[ : 0] (we assume it is well-defined). Then the results above should be compared with a theorem due to K. Brown [B2]: X "\() \() = \(C(fl)) : (fl)6=(1) fl finiteorder Brown's result can be considered as the generalization of the "class equati* *on" from finite group theory.Our result is an extension of the formula for dimCR(G)* *C. x4. EXAMPLES AND A LOCAL VERSION Assume we have a group fitting into an extension 1! 0 ! ! G ! 1as described before. Then will contain a subgroup (p)of finite index normalizing 0 and with (p)=0= Sylp(G). Hence the class of groupsto which the exact formula 3.3 can be applied is a large one. Here we concentrate on familiar examples. Example 4.1: Let = K N H, the amalgamated product of two finite groups over a common subgroup. In this case the group is virtually free [S] and we may take 0to be a free group of finite index. Here X can be taken to be a tree,X=0 a finite graph on which G = =0acts with orbit space N ffl ffl 12 i.e. there are two orbits of vertices, with stabilizers K, H respectively, and * *one orbit of edges, with stabilizer N .In this case we use a spectral sequence due to Seg* *al [Se] for computing equivariantcohomology (indeed it is a very simple Mayer-Viaetoris sequence in this case) to obtain the exact sequence: 0 ! K0G(X=0) ! R(K) R(H) ! R(N ) ! K1G(X=0) ! 0: This may be used to compute KG(X=0). Compare this to the formula [B] ! ! ! \() = !1jKj+!1jHj !1jNj: This applies to = SL2(Z) = Z=6 Z=2Z=4 to yield ae dimCKG (X=0) C = 0;8;ifi*fodd;* even. A similar formula can obviously be proved for any virtually free group of f* *inite vcd, taking into account the edge and vertex stabilizers of the corresponding t* *ree. There is a local way of analyzing K-theory,by using p-adic K-theory, Kp (see [H]). The field C is replaced by Cp,the completion of the algebraic closure of * *Qp, and instead of characters on G we specialize to class functions on Torsp(G) = fg 2 G j g is of order pn, for some n 0g : Then Atiyah's result can be reformulated as Kp(BG) Cp= Cp-valued class functions on Torsp(G): In this setting we obtain a local version of our main result: Theorem 4.2: M (i) Kp(B) Cp = Kp(B0" C (fl))Hfl Cp fl2T(flo)rs p() X (ii) \(Kp(B) Cp) = "\(C(fl)) fl2(fl)Tors 13 == Example 4.3 : Now let = SL3(Z), 0= (3), G = SL3(F3). Note jGj = 243313 but that has no 13-torsion. Hence we deduce M Kp(B) C = K (BC (fl) " (3))Hfl Cp forp = 2; 3: fl 2(fl)T ors p() In particular for p = 3, we see that up to conjugacy, 0 1 0 1 0 0 1 0 1 0 @ 1 0 0A and @ 1 1 0 A 0 1 0 0 0 1 are the only elements of finite order in which are in Tor sp(). Their centrali* *zers are cyclic of order 3 and cyclic of order 6 respectively. We obtain K3(B) C3 = K3(X=) C3 [K3() C3]2: However, from the work of Soule [So], X= is homotopically trivial, hence K3(BSL 3(Z)) C3= [K3() C3]3: Similarly, we have that the centralizers of elements of order 2 or 4 in SL3* *(Z) are rationally acyclic, hence K2(BSL (Z)) C2 is of rank equal to the number of distinct conjugacy classes of elements in SL 3(Z) of order 2 or4. Example 4.4 Let k be a totally real number field with ring of integers O, and let ik denote the Dedekind zetafunction associated to k. The centralizer of eve* *ry finite subgroup in = SL2(O) is finite, except for 1. Let [Torsp()] denote the number of conjugacy classes of non-trivial elements in Torsp(). Then,for any p 6= 2 14 and X ! \(Kp (BSL 2(O)) Cp) = 2ik(1) + 1 !2jHj+ [Torsp()] (H) where H ranges over -conjugacy classes of maximal finite subgroups. For this formula we use an identity due to K. Brown [B1] for "\(SL 2(O)). Example 4.5: Let = GLp1 (Z), p an odd prime. It is well-known that this group has no subgroups of order p2,and furthermore the number of conjugacy classes of elements of order p in is equal tothe class number of p, Cl(p). The centralize* *r of any such element flwill be isomorphic to the group of units U in Z[i], where ii* *s a primitive p-th root of unity. It is also well known that U splits as a direct p* *roduct < fl > Z(p3)=2 Z=2. Hence we obtain M Kp(BGLp1 (Z)) Cp = Kp(B0)G Cp Kp((S1)(p3)=2 ) Cp Cl(p) where 0is a normal torsion-free subgroup of GLp1 (Z) with finite factor group G. In addition, we obtain that \(Kp(B GLp1 (Z)) Cp) = "\(GLp1 (Z)): We point out that using results due to Ash [A], formulae of this type can be ca* *lcu- lated for GLn(Z), provided that p 1 n 2p 3: REFERENCES [A] A. Ash, "Farrell Cohomology of GLn(Z)," to appear in Israel Journal of Mathematics. [A-S1] M. F.Atiyah and G. B. Segal, "Equivariant K-Theory and Completion," Journal of Differential Geometry 3 (1969) 1-18. [A-S2] M. F.Atiyah and G. B. Segal, unpublished. [B] K. Brown, "Cohomology of Groups," Springer-Verlag GTM 87 (1982). [B1] K. Brown, "Euler Characteristics of Discrete Groups and G-Spaces," Inv. 15 [B2] K. Brown, "Complete Euler Characteristics and Fixed-Point Theory," J. Pure& Applied Algebra 24 (1982) 103-121. [B3] K. Brown, "High-dimensional Cohomology of Discrete Groups," Proc. Natl. Acad.Sci. (USA) Vol. 73, No. 6 1795-1797 (1976). [E-G] S. Eilenberg and T. Ganea, "On the Lusternik-Schnirelmann Category of Abstract Groups," Ann. Math. 65 (1957) 517-518. [H] M. Hopkins, "Characters and Elliptic Cohomology," Advances in Homotopy Theory, Salamon, Steer, and Sutherland (editors),LMS Lecture Note Series 139,Cambridge University Press, 1989. [K] N. Kuhn, "Character Rings in Algebraic Topology," Advances in Homotopy Theory, Salamon, Steer, and Sutherland (editors),LMS Lecture Note Series 139,Cambridge University Press, 1989. [H-H] F. Hirzebruch and T. H|fer, "On the Euler Number ofan Orbifold," Math- ematische Annalen 286 (1990), 255-260. [Se] G.B.Segal, "Equivariant K-Theory," Pub. Math. Inst. desHautes Etudes Scient. (Paris), 34 (1968). [S] J-P. Serre, "Cohomologie des Groupes Discrets," Ann. Math. Studies 70 (1971) 77-169. [So] C. Soule, "The Cohomology of SL3(Z)," Topology 17 (1978) 1-22. [tD] T. Tom Dieck, "Transformation Groups and Representation Theory," Lec-