Some groups of finite homological type

Ian J. Leary^{∗}  Müge Saadetoğlu^{†}

June 7, 2005

Abstract For each n ≥ 0 we construct a torsion-free group that satisfies K. S.

Brown’s F HT condition and is F P_{n}, but is not F P_{n+1}.

1 Introduction

While working on comparing different notions of Euler characteristic, K. S. Brown introduced a new homological finiteness condition for discrete groups [5, IX.6]. The group G is said to be of finite homological type or F HT if G has finite virtual cohomological dimension, and for every G-module M whose underlying abelian group is finitely generated, the homology groups H_{i}(G; M) are all finitely generated. If G is F HT, then one may define a ‘naïve Euler characteristic’ for every finite-index subgroup H of G, as the alternating sum of the dimensions of the homology groups of H with rational coefficients.

One question that arises is the connection between F HT and the usual homological finiteness conditions F P and F P_{n}, which were introduced by J.-P. Serre [8]. (We shall define these conditions below.) It is easy to see that any group G of type F P is F HT, and one might conjecture that every torsion-free group that is F HT is also of type F P. The aim of this paper is to show that this is not the case. For each n ≥ 0, we exhibit a torsion-free group G_{n} that is F HT and of type F P_{n}, but that is not of type F P_{n+1}.

Our construction is based on R. Bieri’s construction of a group that is F P_{n} but not F P_{n+1} [2, Prop 2.14]. We also use G. Higman’s group H which has the properties that it has no non-trivial finite quotients and that it is acyclic (i.e., has the same integral homology as the trivial group) [1, 6]. The group G_{0} is just an infinite free product of copies of H, and for n > 0, G_{n} may be described as a free product of two groups of type F P, amalgamating a common subgroup isomorphic to G_{n−1}.

∗Partially supported by NSF grant DMS-0505471

†Supported by the British Council and by the Ohio State Mathematical Research Institute

A construction of the groups G_{n} was given in the University of Southampton PhD thesis of the second named author [7].

2 Definitions

Let G be a discrete group, let ℤG be the integral group ring of G, and let ℤ stand for the trivial ℤG-module, i.e., the module whose underlying abelian group is infinite cyclic upon which each element of G acts as the identity. Modules will be left modules unless otherwise stated.

We begin by recalling some classical homological finiteness conditions, which were introduced by J.-P. Serre [8], but see also [2, Ch I] and [5, Ch VIII]. A projective resolution P_{∗} for ℤ over ℤG is an exact sequence

⋯ → P_{i} → ⋯ → P_{1} → P_{0} → ℤ → 0

of ℤG-modules in which each P_{i} is projective. The group G is F P_{n} if there exists P_{∗} such that P_{i} is finitely generated for all i ≤ n and is F P_{∞} if there is a P_{∗} in which P_{i} is finitely generated for all i. The group G is of finite cohomological dimension if there exists P_{∗} in which some P_{n} = 0, in which case we may take P_{i} = 0 for all i ≥ n. It can be shown that any group of finite cohomological dimension is torsion-free. The group G is F P if G is of finite cohomological dimension and of type F P_{∞}. If G is F P, then there exists a resolution P_{∗} in which each P_{i} is finitely generated and only finitely many P_{i} are non-zero.

If G contains a finite-index subgroup H which is of finite cohomological dimension, G is said to be of finite virtual cohomological dimension or finite vcd. By an argument due to Serre, any group of finite vcd admits an action with finite stabilizers on a finite-dimensional contractible CW-complex [5, Theorem VIII.11.1].

Remark 1 Since there are many interesting discrete groups that are not virtually torsion-free, one might argue that the condition ‘finite vcd’ is an unnatural one, which should be replaced by the condition ‘admits an action with finite stabilzers on a finite-dimensional contractible space’ whenever possible.

Brown defines a group G to be F HT if G is of finite vcd, and for every right G-module M whose underlying abelian group is finitely-generated, each of the homology groups H_{i}(G, M) is a finitely generated abelian group [5, IX.6]. These homology groups may be computed as the homology of the chain complex

M ⊗_{ℤG} P_{∗}

for any projective resolution P_{∗} for ℤ over ℤG. The homology groups of a chain complex of finitely generated abelian groups are themselves finitely generated. Hence each H_{i}(G, M) is finitely generated whenever G is type F P_{∞}, and we see that any group G of finite vcd that is of type F P_{∞} is F HT. In particular, any torsion-free group of type F P is F HT.

Remark 2 In his original papers on Euler characteristics, Brown gave a different definition of finite homological type, which we shall call F HT′. The group G is is F HT′ if G is of finite vcd, and for each torsion-free finiteindex subgroup H ≤ G, the integral homology groups H_{i}(H; ℤ) are finitely generated [3, 4]. By Shapiro’s lemma, H_{i}(H; ℤ) ≅ H_{i}(G; ℤ[H\G]), where ℤ[H\G] denotes the permutation module with basis the right cosets of H in G. Hence any group that is F HT is also F HT′. We do not know whether F HT is equivalent to F HT′.

3 Background

Here we collect together some known results that are used in our construction.

Proposition 3 Let A be a finitely generated abelian group. There exists n such that G = Aut(A) is isomorphic to a subgroup of GL_{n}(ℤ).

Proof. Let T (A) be the torsion subgroup of A, and let A′ be a complement to T (A), so that A = T (A) ⊕ A′ and A′ ≅ ℤ^{r} for some r ≥ 0. The subgroup T (A) is characteristic in A and so there is a natural map

ϕ : G → Aut(T (A)) ⊕ Aut(A/T (A)).

Let H be the subgroup of G consisting of those f ∈ G such that f(A′) = A′. Then H is a direct product

H = Aut(T (A)) ⊕ Aut(A′) ≅ Aut(T (A)) ⊕ GL_{r}(ℤ),

and ϕ induces an isomorphism from H to Aut(T (A)) ⊕ Aut(A/T (A)). Elements f ∈ ker(ϕ) act as the identity on T (A), and for each a ∈ A′, f(a) = a + b for some b ∈ T (A). It follows that ker(ϕ) is isomorphic to Hom(A′, T (A)) ≅ T (A)^{r}, and so ker(ϕ) is a finite group. Since ϕ restricted to H is an isomorphism, it follows that the index of H in G is equal to the order of ker(ϕ), and so the index of H in G is finite.

The finite group Aut(T (A)) is isomorphic to a subgroup of GL_{s}(ℤ) for some s (for example, s = |Aut(T (A))| will suffice). Hence H is isomorphic to a subgroup of GL_{r+s}(ℤ). Equivalently, there is a faithful H-module N whose underlying abelian group is free abelian of rank r + s. If the index of H in G is m, then the induced module ℤG ⊗_{ℤH} N is a faithful G-module whose underlying abelian group is free abelian of rank n = m(r + s). The action map for this module is an embedding of G in GL_{n}(ℤ).

Lemma 4 Let X be a connected CW-complex, let Y be a connected subcomplex, and let y_{0} ∈ Y be a basepoint for both spaces. Let G be π_{1}(X, y_{0}), the fundamental group of X, let i : π_{1}(Y, y_{0}) → G be the induced map of fundamental groups, and let H be a subgroup of G. Let X̂ be the covering space of X with fundamental group H and let Ŷ be the subspace of X̂ consisting of lifts of points of Y. There is a bijective correspondence between components of Ŷ and orbits in the coset space G/H for the action of π_{1}(Y). The fundamental group of the component corresponding to the orbit of the coset gH is a conjugate of i^{−1}(gHg^{−1}) in π_{1}(Y).

Proof. Let X̃ denote the universal cover of X, and let Ỹ denote the subspace corresponding to Y. Pick x_{o} ∈ X̃ a lift of y_{0}. Each component of Ỹ contains some g.x_{0}. A loop γ in Y based at y_{0} lifts to a path from g.x_{0} to g′.x_{0}, for g′ = i([γ]).g, where [γ] denotes the element of π_{1}(Y) represented by the loop γ. The points g.x_{0} and g′.x_{0} map to the same point of X̂ if and only if gH = g′H. Hence there is a path in Ŷ from the image of g.x_{0} to the image of g′.x_{0} if and only if the cosets gH and g′H are in the same π_{1}(Y)-orbit, as claimed.

Each component of Ŷ is a covering space of Y, and so once we have chosen a basepoint we may identify its fundamental group with a subgroup of π_{1}(Y). Taking different basepoints changes this subgroup by conjugation. The image of the point g.x_{0} in Ŷ depends only on the coset gH. If we take as basepoint for a component of Ŷ the image of g.x_{0}, then a loop γ in Y lifts to a loop in Ŷ based at g.x_{0} if and only if the cosets gH and i([γ])gH are equal, or equivalently if and only if i([γ]) ∈ gHg^{−1}.

Corollary 5 Suppose that a group G is expressed as a free product with amalgamation, G = H ∗_{L} K, and that ϕ : G → Q is such that ϕ : L → Q is surjective. Then ker(ϕ) is equal to the free product with amalgamation H′ ∗_{L′} K′, where H′ = ker(ϕ) ∩ H, L′ = ker(ϕ) ∩ L and K′ = ker(ϕ) ∩ K.

Proof. A model for the classifying space BG can be made by joining copies of BH and BK by a cylinder BL×I, where I denotes the unit interval. Take this space to be the space X in Lemma 4, and for X̂ take the regular cover with fundamental group ker(ϕ), so that X̂ is the classifying space for ker(ϕ). Lemma 4 can be applied in the cases Y = BH, Y = BK and Y = BL × I. In each case, it follows that Ŷ is connected, and the fundamental group of Ŷ is H′, K′ or L′ respectively. Hence X̂ is built by joining a copy of BH′ and a copy of BK′ via a cylinder BL′ × I, and so ker(ϕ) ≅ H′ ∗_{L′} K′.

Corollary 6 Suppose that G = H ∗K, and define a homomorphism ϕ : G → K as the identity homomorphism on K and the trivial map on H. The kernel of ϕ is isomorphic to a free product of copies of H indexed by the elements of K.

Proof. Build a classifying space BG as the one-point union BH ∨ BK, and apply Lemma 4 with X̂ being the regular cover corresponding to ker(ϕ). In the case when Y = BK, we see that Ŷ is the universal covering space EK of BK, and in the case when Y = BH, we see that Ŷ is a disjoint union of copies of BH indexed by the elements of K. Hence B(ker(ϕ)) can be constructed by attaching copies of BH indexed by the elements of K to the contractible space EK.

The following theorem is a special case of [2, prop. 2.13(a)].

Theorem 7 (R. Bieri) Let G = H ∗_{L} K be a free product with amalgamation, and suppose that both H and K are F P. Then for any n ≥ 1, G is F P_{n} if and only if L is F P_{n−1}.

The group H below was introduced by Higman, who proved that H has the ‘group theoretic’ properties given in the following theorem [6]. The proof that H has the stated ‘homological properties’ was given by Baumslag, Dyer and Heller in [1], where the group H played an important role in their strengthened version of the Kan-Thurston theorem.

Theorem 8 (G. Higman, G. Baumslag, E. Dyer, A. Heller) Let H be the group defined by the presentation

H = ⟨ a, b, c, d : a^{b} = a^{2}, b^{c} = b^{2}, c^{d} = c^{2}, d^{a} = d^{2} ⟩.

Then H is an infinite torsion-free group, the presentation 2-complex for the above presentation is a classifying space for H, and H admits no non-trivial quotient in which the images of the generators have finite order.

Corollary 9 H as above is a non-trivial torsion-free acyclic group with no proper finite-index subgroups.

4 The groups

Lemma 10 Let M be a module for Higman’s group H whose underlying abelian group is finitely generated. Then H acts trivially on M.

Proof. Let G = Aut(M), the group of abelian group automorphisms of M. An H-module structure on M is a homomomorphism H → G. By Proposition 3, G is isomorphic to a subgroup of GL_{n}(ℤ) for some n. Thus it suffices to show that there are no non-trivial homomorphisms ϕ : H → GL_{n}(ℤ). For each m > 1, let π_{m} : GL_{n}(ℤ) → GL_{n}(ℤ/mℤ) denote the homomorphism ‘reduction modulo m’. By Corollary 9, H has no proper finite-index subgroups, and so the homomorphism

π_{m} ∘ ϕ : H → GL_{n}(ℤ/mℤ)

must be trivial for each m > 1. However, the only matrix in the kernel of all of the π_{m} is the identity matrix, and so ϕ must be the trivial homomorphism.

Remark 11 For any group G, and any right G-module A, a left G-action on A may be defined by g ∗ a = ag^{−1}. This gives a bijection between the left and right G-module structures on any fixed abelian group A.

Proposition 12 Let G_{0} be an infinite free product of copies of Higman’s group H, and let M be a right G_{0}-module whose underlying abelian group is finitely generated. Then G_{0} acts trivially on M, and

H_{0}(G_{0}; M) ≅ M, H_{i}(G_{0}; M) = 0 for i > 0.

Proof. Let M be as in the statement. By Lemma 10 and the remark following it, each copy of H inside G_{0} must act trivially on M. It follows that G_{0} acts trivially on M. Since H is acyclic, it follows that G_{0} is also acyclic, and so the homology of G_{0} with integer coefficients is isomorphic to the integral homology of the trivial group. The universal coefficient theorem allows one to compute the homology of G_{0} with coefficients in any trivial module. Since each H_{i}(G_{0}; ℤ) is free, the tor-term in the universal coefficient theorem vanishes, and so H_{i}(G_{0}; M) ≅ M ⊗_{ℤ} H_{i}(G_{0}; ℤ) for all i, giving the result claimed above.

Corollary 13 The group G_{0} as described above is F HT, is F P_{0} and is not F P_{1}.

Proof. There is a 2-dimensional BG_{0} (consisting of the one point union of infinitely many copies of a 2-dimensional BH), so G_{0} has cohomological dimension at most 2. (In fact, since G_{0} is not free its cohomological dimension is exactly 2, but we do not need this fact.) By Proposition 12, the homology groups H_{i}(G_{0}; M) are all finitely generated whenever M is a right G_{0}-module whose underlying abelian group is finitely generated. A group is F P_{1} if and only if it is finitely generated, and so G_{0} is not F P_{1}. Every group is F P_{0}.

To construct the rest of our examples, we will start by embedding G_{0} in a group of type F P. Let J_{0} be the free product H∗ℤ, and define ϕ_{0} : J_{0} → ℤ by the identity map on ℤ and the trivial map from H to ℤ. Applying Corollary 6, we see that ker(ϕ_{0}) is isomorphic to a free product of infinitely many copies of the Higman group H. From now on, we shall identify G_{0} with ker(ϕ_{0}) ≤ J_{0}.

Let 𝔽_{2} denote the free group on two generators, and let ψ : 𝔽_{2} → ℤ be the homomorphism that sends each of the two generators to 1 ∈ ℤ.

Now suppose that we have already defined a group J_{n} and a homomorphism ϕ_{n} : J_{n} → ℤ. Define a new group J_{n+1} containing J_{n} as a direct factor, and a new homomorphism ϕ_{n+1} : J_{n+1} → ℤ extending ϕ_{n} by

J_{n+1} = J_{n} × 𝔽_{2}, ϕ_{n+1}(g, h) = ϕ_{n}(g) + ψ(h) for all g ∈ J_{n} and h ∈ 𝔽_{2}.

For each n, we shall identify J_{n} with J_{n} × {0} ≤ J_{n+1}. For n > 0, define G_{n} = ker(ϕ_{n} : J_{n} → ℤ).

Proposition 14 For each n ≥ 0, G_{0} is a normal subgroup of G_{n}, and G_{n}/G_{0} ≅ (𝔽_{2})^{n}. For each n ≥ 0, there is an isomorphism G_{n+1} ≅ J_{n} ∗_{G_{n}} J_{n}.

Proof. J_{0} is a direct factor of J_{n}, and G_{0} is a normal subgroup of J_{0}. It follows that G_{0} is normal in J_{n} and that

J_{n}/G_{0} ≅ (J_{0}/G_{0}) × 𝔽_{2}^{n} ≅ ℤ × 𝔽_{2}^{n}.

G_{0} is contained in G_{n} = ker(ϕ_{n}), and so there is an induced homomorphism ϕ̄_{n} : J_{n}/G_{0} → ℤ. Under the above isomorphism J_{n}/G_{0} ≅ ℤ × 𝔽_{2}^{n}, the homomorphism ϕ̄_{n} corresponds to the homomorphism which sends (r, s_{1}, …, s_{n}) ∈ ℤ×𝔽_{2}^{n} to r+ψ(s_{1})+⋯+ψ(s_{n}). Since this map restricts to ℤ×{e}^{n} ≤ ℤ×𝔽_{2}^{n} as the identity map of ℤ, it follows that ker(ϕ̄_{n}) is isomorphic to 𝔽_{2}^{n}. Hence

G_{n}/G_{0} = ker(ϕ̄_{n}) ≅ 𝔽_{2}^{n},

as claimed.

We may write 𝔽_{2} = ℤ ∗ ℤ, and thus we may write

J_{n+1} = J_{n} × (ℤ ∗ ℤ) = (J_{n} × ℤ) ∗_{J_{n}} (J_{n} × ℤ).

Let ϕ′ be the restriction of ϕ_{n+1} to one of the two copies of J_{n}×ℤ. The map ϕ′ is given by the formula ϕ′(g, r) = ϕ_{n}(g)+r. In particular, the restriction of ϕ′ to the ℤ direct factor is the identity, and it follows that ker(ϕ′) is isomorphic to J_{n}. The isomorphism between G_{n+1} and J_{n} ∗_{G_{n}} J_{n} follows by applying Corollary 5 to ϕ_{n+1}.

Theorem 15 For each n ≥ 0, the group G_{n} is torsion-free, is F HT and is F P_{n}, but is not F P_{n+1}.

Proof. The group J_{n} has a finite n + 2-dimensional classifying space, so J_{n} is F P. Also G_{n} (as a subgroup of J_{n}) must have finite cohomological dimension, and so must be torsion-free. (In fact the cohomological dimensions of J_{n} and G_{n} are both equal to n+2.) The group G_{0} is F P_{0} but not F P_{1}. The assertion that G_{n} is F P_{n} but not F P_{n+1} follows by induction, using Bieri’s theorem (Theorem 7) and the description G_{n+1} ≅ J_{n} ∗_{G_{n}} J_{n}.

It remains to check that whenever M is a right G_{n}-module whose underlying abelian group is finitely generated, then each H_{i}(G_{n}; M) is finitely generated. For this we use the Lyndon-Hochschild-Serre (or LHS) spectral sequence for the group extension G_{0} → G_{n} → G_{n}/G_{0}. Let M be a right G_{n}- module whose underlying abelian group is finitely generated. The E^{2}-page of the LHS-spectral sequence has

E_{i,j}^{2} = H_{i}(G_{n}/G_{0}; H_{j}(G_{0}; M)).

By Lemma 12, the subgroup G_{0} acts trivially on M, and H_{0}(G_{0}; M) = M, H_{j}(G_{0}; M) = 0 for j > 0. Also G_{n}/G_{0} ≅ 𝔽_{2}^{n} is a group of type F P. Since the spectral sequence has E_{i,j}^{2} = 0 for j ≠ 0, it must collapse, giving isomorphisms

H_{i}(G_{n}; M) ≅ E_{i,j}^{2} ≅ H_{i}(G_{n}/G_{0}; M) ≅ H_{i}(𝔽_{2}^{n}; M).

Since 𝔽_{2}^{n} is of type F P, it follows that each H_{i}(G_{n}; M) is finitely generated as claimed.

References

[1] G. Baumslag, E. Dyer and A. Heller, The topology of discrete groups, J. Pure Appl. Algebra 16 (1980) 1–47.

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

[3] K. S. Brown, Euler Characteristics of discrete groups and G-spaces, Invent. Math. 27 (1974) 229–264.

[4] K. S. Brown, Euler characteristics of groups: the p-fractional part, Invent. Math. 29 (1975), 1–5.

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

[6] G. Higman, A finitely generated infinite simple group, J. London Math. Soc. 24 (1951) 61–64.

[7] M. Saadetoğlu, Finiteness conditions and Bestvina-Brady groups, PhD Thesis, School of Mathematics, University of Southampton (2005).

[8] J.-P. Serre, Cohomologie des groupes discrets, Prospects in Mathematics, 77–169, Ann. of Math. Studies 70 (1971) Princeton Univ. Press.

Ian Leary: Department of Mathematics, The Ohio State University, 231 West 18th Avenue, Columbus, Ohio 43210-1174, United States. and School of Mathematics, University of Southampton, Southampton, SO17 1BJ, United Kingdom.

   leary@math.ohio-state.edu

Müge Saadetoğlu: School of Mathematics, University of Southampton, Southampton, SO17 1BJ, United Kingdom.

   ms@maths.soton.ac.uk