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 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

⋯ → P_{n+1} → P_{n} → ⋯ → P_{0} → M → 0

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.)

Date: January 5, 2001.

1991 Mathematics Subject Classification. Primary 20J05; Secondary 19J05.

Key words and phrases. finiteness conditions, finite subgroups.

The first author acknowledges support from EPSRC via grant GR/R07813, SFB 478 in Münster, the FIM at the ETH Zürich and the CRM Barcelona.

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;

(b) cd_{ℚ}(G) = n;

(c) G is FP_{n−1} over ℤ.

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.

2. 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

P_{n−1} → P_{n−2} → ⋯ → P_{1} → P_{0} → M → 0

is a partial projective resolution of M in which each P_{i} is finitely generated. Then K_{n}, 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

[M, N] = Hom_{R}(M, N)/P Hom_{R}(M, N).

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

[M, N] → [ΩM, ΩN] → [Ω^{2}M, Ω^{2}N] → ⋯

having the following property, which can be seen by a standard diagramchasing argument. Lemma 2. Suppose [Ω^{i}M, Ω^{i}N] = 0, then [Ω^{j}M, Ω^{j}N] = 0 for all j ≥ i. □

It is now possible to define the Tate cohomology group in degree zero as a direct limit as follows:

Definition.

Êxt_{R}^{0}(M, N) = lim_{→}[Ω^{i}M, Ω^{i}N].

¿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 3. For every i ≥ 0 the following groups are isomorphic:

[K_{i}, K_{i}] ≅ [Ω^{i+1}ℤ, Ω^{i+1}ℤ].

Proof. This follows from Shanuel’s Lemma and an application of the fact that for arbitrary M, N and projective modules P and Q,

[M ⊕ P, N] ≅ [M, N] ≅ [M, N ⊕ Q]

. □ Proof of Proposition 1. Consider a partial projective resolution of ℤ over ℤG where all P_{i}, i ≤ n − 1, are finitely generated:

P_{n−1} → ... → P_{0} → ℤ → 0,

and let K be the kernel of the map P_{n−1} → P_{n−2}. 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:

0 → K ⊗ ℚ → P_{n−1} ⊗ ℚ → ... → P_{0} ⊗ ℚ → ℚ → 0

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 : K ↪ K⊗ℚ the inclusion defined by ι(k) = k ⊗1. Suppose k_{1}, ...k_{s} generate K. For each 1 ≤ j ≤ s there exist λ_{ij} ∈ ℚG such that ιτ(k_{j}) = ∑_{i=1}^{r} λ_{ij}f_{i}. Now pick m ∈ ℤ such that each mλ_{ij} ∈ ℤG. Since τ is a split injection we can precompose the identity id_{K⊗ℚ} = πτ with multiplication by m. Hence the map

K ⟶^{ι} K ⊗ ℚ ⟶^{×m} K ⊗ ℚ

factors through F_{0} and has image in K thus proving the claim. The Claim together with Lemmas 2 and 3 gives that

m[K, K] ≅ m[Ω^{i}ℤ, Ω^{i}ℤ] = 0 for all i ≥ n.

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

mĤ^{0}(H, ℤ) ≅ m[K, K] ≅ lim_{→} m[Ω^{i}ℤ, Ω^{i}ℤ] = 0.

(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 Ĥ^{0}(H, ℤ) ≅ ℤ/|H|ℤ and 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 M of type F P_{n}, which was mentioned at the beginning of the previous section:

P_{n−1} → P_{n−2} → ⋯ → P_{1} → P_{0} → M → 0.

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] 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 4. 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 Q_{1} ≅ Q_{0} ≅ RC. Now suppose that

0 → P_{n} → P_{n−1} → ⋯ → P_{0} → R → 0

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 P_{∗} ⊗ Q_{∗} is a projective resolution of R ⊗ R = R over RG ⊗ RC ≅ R(G × C), of length n + 1. 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 5. 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 ℤGmodule 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 ∏_{j=1}^{n} m_{j}.

Sketch-proof. The above bound is obtained by comparing the two spectral sequences arising from the double complex

E_{0}^{i,j} = Hom_{H}(P_{i}, F_{j}),

where H is a finite subgroup of G and P_{∗} is a complete resolution for H. □

This lemma allows us to prove a slightly weaker version of Proposition 1 using only ordinary Tate cohomology for finite groups.

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 4, 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 − g_{i}). Finally, let

0 → P_{n} → ⋯ → P_{1} → P_{0} → ℚ → 0

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 5 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.

References

[1] D. J. Benson and J. F. Carlson, Products in negative cohomology, J. Pure Appl. Algebra 82 (1992), 107–130.

[2] R. Bieri, Homological Dimension of Discrete Groups, Queen Mary College Mathematics Notes, University of London (1976).

[3] K. S. Brown, Cohomology of Groups, Graduate Texts in Mathematics 87, Springer (1982).

[4] K. S. Brown, Finiteness properties of groups, 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 (1978/79), 254–258.

[6] P. H. Kropholler, On groups of type FP_{∞}, J. Pure Appl. Algebra 90 (1993), 55–67.

[7] I. J. Leary and B. E. A. Nucinkis, Some groups of type VF, preprint, Southampton (2000).

[8] G. Mislin, Wall’s finiteness obstruction, Handbook of algebraic topology, 1259– 1291, North-Holland (1995).

[9] G. Mislin, Tate cohomology for arbitrary groups via satellites, Topology Appl. 56 (1994), 293–300.

Faculty of Mathematical Studies, University of Southampton, Southampton, SO17 1BJ, United Kingdom

E-mail address: I.J.Leary@maths.soton.ac.uk

Department of Mathematics, ETH-Zentrum, 8092 Zürich, Switzerland

E-mail address: bean@math.ethz.ch