Bounding the orders of finite subgroups

Publ. Mat. 45 (2001), 259–264 BOUNDING THE ORDERS OF FINITE SUBGROUPS Ian J. Leary and Brita E. A. Nucinkis Abstract We give homological conditions on groups such that whenever the conditions hold for a group G, there is a bound on the orders of finite subgroups of G. This extends a result of P. H. Kropholler. We also suggest a weaker condition under which the same conclusion might hold. 1. Introduction Let Rbe a non-trivial unital ring. An R-module Mis said to be of type FPnif there is a projective resolution ···→Pn+1 →Pn→···→P0→M→0 of Mover Rin which P0,...,P nare finitely generated. Mis said to be of type FP∞if Mis FPnfor each n. Similarly, Mis said to be of type FP (resp. FL) over Rif there is a resolution of Mof finite length in which each term is a finitely generated projective (resp. free) module. For any discrete group Gand commutative ring R, the augmentation homomorphism RG →Rgives Rthe structure of a module for the group algebra RG. The group Gis said to be FPn(resp. FP∞,FP,FL) over Rif the RG-module Ris FPn(resp. FP∞,FP,FL) in the above sense. The cohomological dimension of Gover R, denoted by cdR(G), is the projective dimension of Ras an RG-module. For further information concerning these definitions, see [2] or Chapter VIII of [3]. As usual, let Qand Zdenote the rational numbers and the integers respectively. We prove the following. Proposition 1. Let Gbe a group with cdQ(G)=n<∞and suppose that Gis of type FPnover Z. Then there is a bound on the orders of finite subgroups of G. 2000 Mathematics Subject Classification. Primary: 20J05; Secondary: 19J05. Key words. Finiteness conditions, finite subgroups. The first author acknowledges support from EPSRC via grant GR/R07813, SFB 478 in M¨unster, the FIM at the ETH Z¨urich and the CRM Barcelona. 260 I. J. Leary, B. E. A. Nucinkis A similar result was proved by P. H. Kropholler in Section 5 of [6], under the extra hypothesis that Gshould be FP∞over Z. His proof made use of the complete cohomology introduced by D. Benson, J. Carlson, G. Mislin and F. Vogel [1], [9] as will ours. (Complete cohomology can be viewed as a generalization of Tate cohomology.) The conclusion does not hold for all groups of type FPn−1over Z. K. S. Brown has shown [4] that for each n>0, the Houghton groups [5] afford an example of a group G=G(n) such that: (a) Gcontains the infinite, finitary symmetric group; (b) cdQ(G)=n; (c) Gis FPn−1over Z. The authors have recently constructed groups Gof type FP∞over Zwith cdQGfinite that contain infinitely many conjugacy classes of finite subgroups [7], and it was these examples that led to the authors’ interest in Proposition 1. It is not known whether there is a bound on the orders of finite subgroups for every Gof type FP over Q. Some remarks concerning this question will be made at the end of the paper. 2. Proofs Before starting, we recall a basic property of FPn-modules. Suppose that Mis an R-module of type FPn, and that Pn−1→Pn−2→···→P1→P0→M→0 is a partial projective resolution of Min which each Piis finitely generated. Then Kn−1, defined as the kernel of the map from Pn−1to Pn−2, is finitely generated. We shall also give a brief outline of Benson and Carlson’s version of generalized Tate cohomology for arbitrary rings R[1]. For R-modules M and Nlet PHomR(M,N) be the group of all R-module homomorphisms which factor through a projective, and let [M,N] = HomR(M,N)/P HomR(M,N). For arbitrary R-modules Mlet FM be the free module on the set M and ΩMis the kernel of the canonical projection FM M. Let ΩiM= Ω(Ωi−1M). Then there is a well defined sequence of maps [M,N]→[ΩM,ΩN]→[Ω2M,Ω2N]→··· and it is now possible to define the Tate cohomology group in degree zero as a direct limit as follows: Bounding the Orders of Finite Subgroups 261 Definition.  Ext0 R(M,N) = lim −→[ΩiM,ΩiN]. From now on we shall concentrate on projective resolutions P∗Z of the trivial module Zover the group-ring ZG. Let Kibe the kernel of the map Pi→Pi−1for i≥1 and K0= ker(P0Z). Lemma 2. For every i≥0the following groups are isomorphic: [Ki,K i]∼ =[Ωi+1Z,Ωi+1Z]. Proof: This follows from Shanuel’s Lemma and an application of the fact that for arbitrary M,Nand projective modules Pand Q, [M⊕P,N]∼ =[M,N]∼ =[M,N ⊕Q]. Proof of Proposition 1: Consider a partial projective resolution of Zover ZGwhere all Pi,i≤n−1, are finitely generated: Pn−1→···→P0→Z→0, and let Kbe the kernel of the map Pn−1→Pn−2.AsGis of type FPn the kernel Kis finitely generated. Since tensoring with Qis exact we obtain a projective resolution of Qover QG, which is of type FP: 0→K⊗Q→Pn−1⊗Q→···→P0⊗Q→Q→0. Therefore K⊗Qis a direct summand of a finite rank QG-free module F, freely generated by {f1,...,f r}, say. Let F0be the free ZG-module on these generators. Claim. There is an integer m, such that multiplication with mfrom K to Kfactors through F0. Let π:FK⊗Qbe the projection onto K⊗Qand τ:K⊗Q→Fbe a splitting, i.e., a map such that πτ =id K⊗Q. Denote by ι:K→K⊗Q the inclusion defined by ι(k)=k⊗1. Suppose k1,...,k sgenerate K. For each 1 ≤j≤sthere exist λij ∈QGsuch that ιτ(kj)=r i=1 λijfi. Now pick m∈Zsuch that each mλij ∈ZG. Since τis a split injection we can precompose the identity idK⊗Q=πτ with multiplication by m. Hence the map Kι −→ K⊗Q×m −→ K⊗Q factors through F0and has image in Kthus proving the claim. The claim together with Lemma 2 gives that m[K, K]∼ =m[ΩnZ,ΩnZ]=0. 262 I. J. Leary, B. E. A. Nucinkis Complete cohomology agrees with ordinary Tate cohomology for finite groups and we can therefore take an arbitrary finite subgroup Hof G and get that m H0(H,Z)∼ =m[K, K]∼ =lim −→ m[ΩiZ,ΩiZ]=0. The direct limit vanishes since for every ϕ∈Hom(ΩiZ,ΩiZ), which factors through a projective, the induced maps Ωjϕ:Ω i+jZ→Ωi+jZalso factor through projectives. (Note that ΩiZhere denotes the ith kernel in the Benson-Carlson construction for ZHand not ZGas earlier used. This does not change the outcome, though.) But also  H0(H,Z)∼ =Z/|H|Zand therefore the group order is a divisor of m, thus bounded. 3. FP-groups over Q Let us consider again the partial resolution of the R-module Mof type FPn, which was mentioned at the beginning of the previous section: Pn−1→Pn−2→···→P1→P0→M→0. There is such a partial resolution in which each Piis finitely generated and free. If also Mhas projective dimension n, then Mis FP.IfMhas projective dimension nand the Piare finitely generated free modules, then Mis FL if and only if Kis stably free. These results can be found in [3, Sections VIII.4–VIII.6]. The following lemma is well-known, but we could not find a reference, so we briefly sketch a proof. A similar topological result appears in [8, Corollary 5.5]. Lemma 3. Let Cdenote an infinite cyclic group. For any R,ifGis a group of type FP over R, then G×Cis of type FL over R. Proof: There is a free resolution Q∗of Rover RC of length one, with Q1∼ =Q0∼ =RC. Now suppose that 0→Pn→Pn−1→···→P0→R→0 is a projective resolution of Rover RG in which each Piis finitely generated, and Piis free for i<n. Let Pbe such that Pn⊕Pis a finitely-generated free RG-module. Writing ⊗for tensor products over R, the total complex T∗for the double complex P∗⊗Q∗is a projective resolution of R⊗R=Rover RG ⊗RC ∼ =R(G×C), of length n+1. Each Tiis finitely generated and Tiis free for i<n. Let S∗be the exact chain complex consisting of one copy of P⊗RC in degree n+1 and one copy in degree n, with the identity map as the boundary. Then S∗⊕T∗ is a finite free resolution of Rover R(G×C). Bounding the Orders of Finite Subgroups 263 Lemma 4. Let Fn→···→F0be a finite-length chain complex of free ZG-modules, suppose that H0(F∗)is isomorphic to the trivial ZG-module Z, and that for each j>0, there exists an integer mj>0such that multiplication by mjannihilates Hj(F∗). Then any finite subgroup of G has order dividing n j=1 mj. Sketch-proof: The above bound is obtained by comparing the two spectral sequences arising from the double complex Ei,j 0= HomH(Pi,F j), where His a finite subgroup of Gand P∗is a complete resolution for H. These lemmas can be used to prove a slightly weaker version of Proposition 1 using only ordinary Tate cohomology for finite groups. Suppose that Gis FPnover Z,FP over Q, and cdQ(G)=n−1. By Lemma 3, G=G×Cis FPnover Z,FL over Q, and cdQ(G)=n. A sequence of free ZG-modules satisfying the conditions of Lemma 4 can then be constructed. Let us now consider the problem of bounding the orders of finite subgroups of an arbitrary group of type FP over Q. Such a Gis finitely generated, and by Lemma 3, we may assume without loss of generality that Gis FL over Q. Let P0be a free QG-module of rank one with generator v, and let P1be QG-free on a set e1,...,e mbijective with a set g1,...,g mof generators for G. Define a map from P0to Qby v→ 1 and a map from P1to P0by ei→ (1 −gi)v. Finally, let 0→Pn→···→P1→P0→Q→0 be a finite free resolution of Qover QGextending this partial resolution. Now let F0(resp. F1) be the ZG-submodule of P0(resp. P1) generated by v(resp. e1,...,e m). For i≥2, if Fi−1has already been chosen, let FibeaZG-lattice in Pi(i.e., a ZG-free ZG-submodule such that Q⊗Fi=Pi), such that the image of Fiin Pi−1is contained in Fi−1. This defines a finite chain complex F∗of finitely-generated free ZG-modules such that H0(F∗)∼ =Zand Hi(F∗) is torsion for i>0. If one could bound the exponent of the torsion in Hi(F∗), Lemma 4 could be applied to bound the orders of finite subgroups of G. Note that in general Hi(F∗) will not be finitely generated as ZG-module. For example, if Gis not FP2over Z, then H1(F∗) will not be finitely generated. Acknowledgement. The authors thank the referee for carefully checking an earlier version of this article. 264 I. J. Leary, B. E. A. Nucinkis References [1] D. J. Benson and J. F. Carlson, Products in negative cohomology, J. Pure Appl. Algebra 82(2) (1992), 107–129. [2] R. Bieri,“Homological dimension of discrete groups”, Queen Mary College Mathematics Notes, Mathematics Department, Queen Mary College, London, 1976. [3] K. S. Brown,“Cohomology of groups”, Graduate Texts in Mathematics 87, Springer-Verlag, New York, 1982. [4] K. S. Brown, Finiteness properties of groups, in: “Proceedings of the Northwestern conference on cohomology of groups” (Evanston, Ill., 1985), J. Pure Appl. Algebra 44 (1987), 45–75. [5] C. H. Houghton, The first cohomology of a group with permutation module coefficients, Arch. Math. (Basel) 31(3) (1978/79), 254–258. [6] P. H. Kropholler, On groups of type (FP)∞,J. Pure Appl. Algebra 90(1) (1993), 55–67. [7] I. J. Leary and B. E. A. Nucinkis,Some groups of type VF, Preprint (2000). [8] G. Mislin, Wall’s finiteness obstruction, in “Handbook of algebraic topology”, North-Holland, Amsterdam, 1995, pp. 1259–1291. [9] G. Mislin, Tate cohomology for arbitrary groups via satellites, Topology Appl. 56(3) (1994), 293–300. Faculty of Mathematical Studies University of Southampton Southampton SO17 1BJ United Kingdom E-mail address:[email protected] Department of Mathematics ETH-Zentrum 8092 Z¨urich Switzerland E-mail address:[email protected] Primera versi´o rebuda el 18 de desembre de 2000, darrera versi´o rebuda el 23 de gener de 2001.