BOUNDING THE ORDERS OF FINITE SUBGROUPS

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.


Introduction
Let R be a non-trivial unital ring.An R-module M is said to be of type FP n if there is a projective resolution of M over R in which P 0 , . . ., P n are finitely generated.M is said to be of type FP ∞ if M is FP n for each n.Similarly, M is said to be of type FP (resp.FL) over R if there is a resolution of M of finite length in which each term is a finitely generated projective (resp.free) module.For any discrete group G and commutative ring R, the augmentation homomorphism RG → R gives R the structure of a module for the group algebra RG.The group G is said to be FP n (resp.FP ∞ , FP , FL) over R if the RG-module R is FP n (resp.FP ∞ , FP , FL) in the above sense.The cohomological dimension of G over R, denoted by cd R (G), is the projective dimension of R as an RG-module.For further information concerning these definitions, see [2] or Chapter VIII of [3].As usual, let Q and Z denote the rational numbers and the integers respectively.We prove the following.

Proposition 1.
Let G be a group with cd Q (G) = n < ∞ and suppose that G is of type FP n over Z. Then there is a bound on the orders of finite subgroups of G.
A similar result was proved by P. H. Kropholler in Section 5 of [6], under the extra hypothesis that G should 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 FP n−1 over 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) G contains the infinite, finitary symmetric group; The authors have recently constructed groups G of type FP ∞ over Z with cd Q G finite 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 G of type FP over Q.Some remarks concerning this question will be made at the end of the paper.

Proofs
Before starting, we recall a basic property of F P n -modules.Suppose that M is an R-module of type FP n , and that is a partial projective resolution of M in which each P i is finitely generated.Then K n−1 , defined as the kernel of the map from P n−1 to P n−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 N let P Hom R (M, N ) be the group of all R-module homomorphisms which factor through a projective, and let For arbitrary R-modules M let F M be the free module on the set M and ΩM is the kernel of the canonical projection F M M .Let Ω i M = Ω(Ω i−1 M ).Then there is a well defined sequence of maps and it is now possible to define the Tate cohomology group in degree zero as a direct limit as follows: From now on we shall concentrate on projective resolutions P * Z of the trivial module Z over the group-ring ZG.Let K i be the kernel of the map P i → P i−1 for i ≥ 1 and K 0 = ker(P 0 Z).

Lemma 2.
For every i ≥ 0 the following groups are isomorphic: Proof: This follows from Shanuel's Lemma and an application of the fact that for arbitrary M , N and projective modules P and Q, Proof of Proposition 1: Consider a partial projective resolution of Z over ZG where all P i , i ≤ n − 1, are finitely generated: and let K be the kernel of the map As G is of type F P n the kernel K is finitely generated.Since tensoring with Q is exact we obtain a projective resolution of Q over QG, which is of type F P : Therefore K ⊗ Q is a direct summand of a finite rank QG-free module F , freely generated by {f 1 , . . ., f r }, say.Let F 0 be the free ZG-module on these generators.

Claim.
There is an integer m, such that multiplication with m from K to K factors through F 0 .
Let π : F K ⊗Q be the projection onto K ⊗Q and τ : K ⊗Q → F be a splitting, i.e., a map such that πτ = id K⊗Q .Denote by ι : Since τ is a split injection we can precompose the identity id K⊗Q = πτ with multiplication by m.Hence the map factors through F 0 and has image in K thus proving the claim.
The claim together with Lemma 2 gives that Complete cohomology agrees with ordinary Tate cohomology for finite groups and we can therefore take an arbitrary finite subgroup H of G and get that The direct limit vanishes since for every ϕ ∈ Hom(Ω i Z, Ω i Z), which factors through a projective, the induced maps Ω j ϕ : Ω i+j Z → Ω i+j Z also factor through projectives.(Note that Ω i Z here denotes the ith kernel in the Benson-Carlson construction for ZH and not ZG as earlier used.This does not change the outcome, though.)But also H 0 (H, Z) ∼ = Z/|H|Z and therefore the group order is a divisor of m, thus bounded.

FP-groups over Q
Let us consider again the partial resolution of the R-module M of type F P n , which was mentioned at the beginning of the previous section: There is such a partial resolution in which each P i is finitely generated and free.If also M has projective dimension n, then M is FP .If M has projective dimension n and the P i are finitely generated free modules, then M is FL if and only if K is stably free.These results can be found in [3,].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 C denote an infinite cyclic group. For any R, if G is a group of type FP over R, then G × C is of type FL over R.
Proof: There is a free resolution Q * of R over RC of length one, with is a projective resolution of R over RG in which each P i is finitely generated, and P i is free for i < n.Let P be such that P n ⊕ P is a finitely-generated free RG-module.Writing ⊗ for tensor products over R, the total complex T * for the double complex Each T i is finitely generated and T i is 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 R over R(G × C).Lemma 4. Let F n → • • • → F 0 be a finite-length chain complex of free ZG-modules, suppose that H 0 (F * ) is isomorphic to the trivial ZG-module Z, and that for each j > 0, there exists an integer m j > 0 such that multiplication by m j annihilates H j (F * ).Then any finite subgroup of G has order dividing n j=1 m j .Sketch-proof: The above bound is obtained by comparing the two spectral sequences arising from the double complex E i,j 0 = Hom H (P i , F j ), where H is a finite subgroup of G and 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 G is FP n over Z, FP over Q, and cd 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 G is finitely generated, and by Lemma 3, we may assume without loss of generality that G is FL over Q.Let P 0 be a free QG-module of rank one with generator v, and let P 1 be QG-free on a set e 1 , . . ., e m bijective with a set g 1 , . . ., g m of generators for G. Define a map from P 0 to Q by v → 1 and a map from P 1 to P 0 by e i → (1 be a finite free resolution of Q over QG extending this partial resolution.Now let F 0 (resp.F 1 ) be the ZG-submodule of P 0 (resp.P 1 ) generated by v (resp.e 1 , . . ., e m ).For i ≥ 2, if F i−1 has already been chosen, let F i be a ZG-lattice in P i (i.e., a ZG-free ZG-submodule such that Q⊗F i = P i ), such that the image of F i in P i−1 is contained in F i−1 .This defines a finite chain complex F * of finitely-generated free ZG-modules such that H 0 (F * ) ∼ = Z and H i (F * ) is torsion for i > 0. If one could bound the exponent of the torsion in H i (F * ), Lemma 4 could be applied to bound the orders of finite subgroups of G.Note that in general H i (F * ) will not be finitely generated as ZG-module.For example, if G is not FP 2 over Z, then H 1 (F * ) will not be finitely generated.