On finite subgroups of groups of type VF

Ian J. Leary^{∗}

June 10, 2005

Abstract

For any finite group Q not of prime power order, we construct a group G that is virtually of type F, contains infinitely many conjugacy classes of subgroups isomorphic to Q, and contains only finitely many conjugacy classes of other finite subgroups.

1 Introduction

A group H is said to be of type F if there is a finite classifying space for H, i.e., if there exists a finite simplicial complex whose fundamental group is isomorphic to H and whose universal cover is contractible. A group of type F is necessarily torsion-free. It is easily seen that any finite-index subgroup of a group of type F is also of type F.

A group G is said to be of type V F if G contains a finite-index subgroup H which is of type F, i.e., if G is virtually of type F. If H has index n in G, then the kernel of the action of G on the cosets of H has index at most n!. Hence any group of type V F contains a finite-index normal subgroup of type F, and so for any group G of type V F there is a bound on the orders of finite subgroups of G.

K. S. Brown’s book ‘Cohomology of Groups’ contains a result that implies that a group of type V F can contain only finitely many conjugacy classes of subgroups of prime power order [4, IX.13.2]. The question of whether a group of type V F could ever contain infinitely many conjugacy classes of finite subgroups was posed in [11, 8], and remained unanswered until Brita Nucinkis and the author constructed examples in [7]. These examples may be summarized as follows:

∗partially supported by NSF grant DMS-0505471

Theorem 1 Let Q be a finite group admitting a simplicial action on a finite contractible simplicial complex L such that the fixed point set L^{Q} is empty. Then there is a group H_{L} of type F (depending only on L) and an action of Q on H_{L} such that the semi-direct product H_{L}:Q contains infinitely many conjugacy classes of subgroup isomorphic to Q.

R. Oliver has shown that a finite group Q admits an action on a finite contractible L without a global fixed point if and only if Q is not expressible as p-group-by-cyclic-by-q-group for any primes p and q [9]. (Oliver’s main result is the construction of actions: the proof that actions do not exist in the other cases is far simpler and we include it in Section 3.)

The purpose of this paper is to close the gap between Brown’s result and the construction of Theorem 1. For any finite group Q that is not of prime power order, we construct a group H of type F with an action of Q so that the semi-direct product H:Q contains infinitely many conjugacy classes of subgroup isomorphic to Q, and finitely many conjugacy classes of other finite subgroups. As a corollary we obtain the following apparently stronger result.

Theorem 2 Let Q = {Q_{1}, …, Q_{n}} be a finite list of isomorphism types of finite group, such that no Q_{i} is a group of prime power order. There exists a group G = G(Q) of type V F such that G contains infinitely many conjugacy classes of subgroup isomorphic to a finite group Q if and only if Q ∈ Q.

In particular, it follows that a group of type V F may contain infinitely many conjugacy classes of elements of finite order, although any such group can only contain finitely many conjugacy classes of elements of prime power order.

Our techniques also apply to other weaker finiteness conditions. Recall that a group G is of type F P over a ring R if the trivial module R for the group ring RG admits a finite resolution by finitely generated projective RGmodules, i.e., if and only if there is an integer n and an exact sequence of RG-modules

0 → P_{n} → ⋯ → P_{1} → P_{0} → R → 0

in which each P_{i} is a finitely generated projective. If there exists such a sequence in which each P_{i} is a finitely generated free module, then G is said to be of type F L over R.

In [7] Brita Nucinkis and the author proved the following.

Theorem 3 Let Q be a finite group admitting a simplicial action on a finite ℚ-acyclic simplicial complex L such that the fixed point set L^{Q} is empty. Then there is a virtually torsion-free group G = H_{L}:Q of type F P over ℚ containing infinitely many conjugacy classes of subgroup isomorphic to Q.

R. Oliver has shown that a finite group Q admits such an action if and only if Q is not of the form cyclic-by-p-group for some prime p [9]. In particular, the above construction did not give rise to any groups of type F P over ℚ containing infinitely many conjugacy classes of elements of finite order. The question of whether such groups can exist was posed by H. Bass in [1, 11]. One reason why this question is of interest is that if G contains infinitely many conjugacy classes of elements of finite order, then the Grothendieck group K_{0}(ℚG) may be shown to have infinite rank. (We give a proof of this fact below in Theorem 26.)

Any group of type F is of type F P over any ring R, and a group G of type V F is of type F P over any ring R in which the orders of all finite subgroups of G are units. In particular, every group of type V F is of type F P over ℚ. It follows that examples coming from Theorem 2 may be used to answer Bass’s question. By Brown’s result, groups of type V F necessarily contain only finitely many conjugacy classes of elements of prime power order. This is not the case for groups of type F P over ℚ, and in fact for any non-trivial finite group Q we construct a group of type F P over ℚ containing infinitely many conjugacy classes of subgroup isomorphic to Q, and finitely many conjugacy classes of other finite subgroup. The following is a corollary of our result.

Theorem 4 Let Q = {Q_{1}, …, Q_{n}} be a finite list of isomorphism types of non-trivial finite groups. There exists a virtually torsion-free group G = G(Q) of type F P over ℚ such that G contains infinitely many conjugacy classes of subgroup isomorphic to a finite group Q if and only if Q ∈ Q.

The groups H_{L} appearing in the statements of Theorems 1 and 3 are the groups introduced by M. Bestvina and N. Brady, who used them to solve a number of open problems concerning homological finiteness conditions [2]. In particular, in the case when L is a finite acyclic complex that is not contractible, they showed that the group H_{L} is of type F L over ℤ but is not finitely presented. The main idea in [7] was to allow a finite group Q to act on the complex L, and hence on the group H_{L}.

The main idea in this paper is to consider Bestvina-Brady groups H_{L} for infinite complexes L. If Q is any finite group not of prime power order, then there exists a complex L with a ℤ × Q-action such that

1. L is contractible;

2. ℤ × Q acts cocompactly on L;

3. all cell stabilizers are finite;

4. {0} × Q fixes no point of L. The first three properties together imply that the semi-direct product H_{L}:ℤ is of type F, and the fourth property implies that the semi-direct product H_{L}:(ℤ×Q) contains infinitely many conjugacy classes of subgroup isomorphic to Q. A construction for L as above in the case when Q is cyclic was given by Conner and Floyd [5]. In Section 3 we give a construction for arbitrary finite Q which was shown to us by Bob Oliver.

A similar (but simpler) construction involving an infinite ℚ-acyclic complex L is used in proving our theorem concerning groups of type F P over ℚ.

In the final section of the paper we discuss some further finiteness properties of the groups that we construct. We show that the groups are residually finite, although we are unable to decide whether they are linear. We also show that each of the groups used in the proofs of Theorems 2 and 4 occurs as the kernel of a map to ℤ from a group that acts cocompactly with finite stabilizers on a CAT(0) cube complex. Acknowledgements The work in this paper builds on the author’s joint work with Brita Nucinkis and uses theorems concerning actions of finite groups which the author learned from Bob Oliver. The author gratefully acknowledges their contributions to this work. Some of this work was done at Paris 13 and at the ETH, Zürich. The author thanks these institutions for their hospitality.

2 Bestvina-Brady groups

In this section we define the Bestvina-Brady group H_{L} associated to a flag complex L, and we check that some of the results in [2, 7] extend to the case when L is an infinite flag complex.

A flag complex, L, is a simplicial complex which contains as many higher dimensional simplices as possible, given its 1-skeleton. In other words, whenever the complete graph on a finite subset of the vertex set of L is contained in the 1-skeleton of L, then there is a simplex of L with that set of vertices. The realisation of any poset is a flag complex (since a subset is totally ordered if any two of its members are comparable). In particular, the barycentric subdivision of any simplicial complex is a flag complex.

Given a flag complex L, the associated right-angled Artin group G_{L} is the group with generators the vertices of L subject only to the relations that the ends of each edge commute. There is a model for the classifying space BG_{L} with one n-dimensional cubical cell for each (n − 1)-simplex of L (including one vertex corresponding to the empty simplex in L). Let X_{L} denote the universal cover of this space. Cells of X_{L} are n-cubes of the form (g, v_{1}, …, v_{n}) where (v_{1}, …, v_{n}) is an n − 1-simplex of L and g is an element of G. The ith pair of opposite faces of this n-cube consists of the cubes (g, v_{1}, …, v̂_{i}, …, v_{n}) and (gv_{i}, v_{1}, …, v̂_{i}, …, v_{n}), where gv_{i} is the product of two elements of G_{L}, and as usual v̂_{i} means ‘omit v_{i}’. The action of G_{L} is given by g′(g, v_{1}, …, v_{n}) = (g′g, v_{1}, …, v_{n}). If σ = (v_{1}, …, v_{n}) is a simplex of L, we will write (g, σ) in place of (g, v_{1} …, v_{n}). In particular, we will write (g) for a vertex of X_{L}.

The space X_{L} admits the structure of a CAT(0) cubical complex: there is a geodesic CAT(0) metric on X_{L} in which each cubical cell is isometric to a standard Euclidean unit cube, and the action of G_{L} is by isometries of this metric. In the case when L is infinite, X_{L} is not locally finite, and the metric topology on X_{L} is not the same as the CW-topology, but this will not cause any difficulties.

Suppose now that f : L → L′ is a simplicial map. Then f defines a group homomorphism f_{∗} : G_{L} → G_{L′}, which takes the generator v to the generator f(v), and f defines a piecewise-linear continuous map f_{!} : X_{L} → X_{L′}, which takes the vertex (g) to the vertex (f(g)), and extends linearly across each cube. The map f_{!} is G_{L}-equivariant, where f_{∗} is used to define the G_{L}- action on X_{L′}, and so induces a map from X_{L}/G_{L} to X_{L′}/G_{L′}, which is an explicit construction for the map B(f_{∗}) : BG_{L} → BG_{L′}. If f is an injective simplicial map, then f_{∗} is an injective group homomorphism and f_{!} is an isometric embedding.

Two special cases of this construction are of interest to us. Firstly, any group Γ of automorphisms of L gives rise to a group of automorphisms of G_{L} and to a group of cellular automorphisms of X_{L}/G_{L}. Since the unique vertex in X_{L}/G_{L} is fixed by Γ, the group of all lifts of elements of Γ to the covering space X_{L} → X_{L}/G_{L} is the semi-direct product G_{L}:Γ, where Γ acts on G_{L} via the action described above.

Secondly, let ∗ denote a 1-point simplicial complex. For this choice of simplicial complex, G_{∗} is infinite cyclic, and X_{∗} is the real line triangulated with one orbit of vertices and one orbit of edges. For any L, there is a unique map f_{L} : L → ∗, and the Bestvina-Brady group H_{L} is defined to be the kernel of f_{L∗} : G_{L} → G_{∗}. The map f_{L!} : X_{L} → X_{∗} ≅ ℝ may be considered as defining a ‘height function’ on X_{∗}. Identifying the integers ℤ ⊆ ℝ with the vertex set in X_{∗}, one sees that f_{L!} sends each vertex of X_{L} to an integer, and that each cube of X_{L} has a unique minimal and maximal vertex for this height function. For the cube C, we shall write min(C) and max(C) respectively for its minimal and maximal vertices. Any simplicial map f : L → L′ fits in to a commutative triangle with f_{L} : L → ∗ and f_{L′} : L′ → ∗, and hence one obtains an induced map f_{∗} : H_{L} → H_{L′}. In particular, if Γ is a group of simplicial automorphisms of L, then the semi-direct product H_{L}:Γ is defined and is equal to the kernel of the composite G_{L}:Γ → G_{∗} × Γ → G_{∗}.

The work of Bestvina and Brady [2] relies on a study of the height function f : X_{L} → X_{∗} = ℝ. We recall part of this, and check that it applies to the case when L is infinite (which was not considered in [2]).

Pick a point c in the interior of an edge of X_{∗}, and define Y = Y_{L} = f^{−1}(c) ⊆ X_{L}. (The point c will remain fixed for the remainder of this section, but will be suppressed from the notation.) Give Y the structure of a polyhedral CW-complex by taking as cells the sets C ∩ Y where C is a cube of X_{L}. Note that the CW-structure on Y gives rise to the same topology as the subspace topology coming from the CW-topology on X.

Now let C be a cube in X_{L} whose highest vertex is v_{1}. Define a subset C_{c} of C by

C_{c} = C ∩ f^{−1}([c, ∞)) = C ∩ f^{−1}([c, f(v_{1})]).

Similarly, if the lowest vertex of C is v_{0}, define a subset C^{c} by

C^{c} = C ∩ f^{−1}((−∞, c]) = C ∩ f^{−1}([f(v_{0}), c]).

If C = (g, σ) for some simplex σ ∈ L, then the link of v_{1} in C is homeomorphic to σ. It follows that if f(v_{1}) > c, then C_{c} is homeomorphic to the cone on L with vertex v_{1}. If f(v_{1}) < c, then C_{c} is empty. Similarly, if f(v_{0}) < c then C^{c} is empty, and otherwise C^{c} is homeomorphic to the cone on σ. Now for v a vertex of X_{L}, define F (v) to be either

F (v) = \begin{cases}⋃_{v=max(C)} C_{c} & f(v) > c \\ ⋃_{v=min(C)} C^{c} & f(v) < c \end{cases}

For each v, one may show that F (v) is homeomorphic to the cone on L with vertex v. (Here, as usual, we are using the CW-topology on both F (v) and the cone on L.) Now for a, b ∈ X_{∗} = ℝ with a < c < b, define a subspace Y_{[a,b]} of X_{L} by

Y_{[a,b]} = Y ∪ ⋃_{a≤f(v)≤b} F (v).

Each Y_{[a,b]} is a CW-complex, with cells the truncated cubes C_{c}, C^{c} and C ∩ Y for each cube C of X_{L}, and if α ≤ a < c < b ≤ β, then Y_{[a,b]} is a subcomplex of Y_{[α,β]}. As a decreases (resp. b increases) the complex Y_{[a,b]} only changes as a (resp. b) passes through an integer. For each a < c < b, one has that Y_{[a−1,b+1]} is homeomorphic to Y_{[a,b]} with a family of subspaces homeomorphic to L coned off. (There is one such cone for each vertex in Y_{[a−1,b+1]} − Y_{[a,b]}.) Thus one obtains the following lemma and corollary due to Bestvina-Brady [2], for any simplicial complex L.

Lemma 5 If L is contractible, then for any a < c < b, the inclusion of Y in Y_{[a,b]} is a homotopy equivalence. If L is R-acyclic for some ring R, then for any a < c < b, the inclusion of Y in Y_{[a,b]} induces an isomorphism of R-homology.

Corollary 6 If L is contractible, then Y is contractible. If L is R-acyclic, then Y is R-acyclic.

Proof. We know that X_{L} is contractible, and the lemma implies that the inclusion Y → X_{L} is a homotopy equivalence if L is contractible and is an R-homology isomorphism if L is R-acyclic.

Theorem 7 Suppose that Γ acts freely cocompactly on a simplicial complex L. If L is contractible, then H_{L}:Γ is type F. If L is R-acyclic, then H_{L}:Γ is type F L over R.

Proof. It follows from Corollary 6 that Y is contractible or R-acyclic whenever L is. Thus it suffices to show that H_{L}:Γ acts freely cocompactly on Y. To see this, first note that G_{L}:Γ has only finitely many orbits of cells in its action on X_{L}. If C is an n-cube of X_{L} with top vertex v, then C ∩ Y is non-empty if and only if c < f(v) < c + n. It follows that each G_{L}:Γ-orbit of n-cubes in X_{L} gives rise to exactly n H_{L}:Γ-orbits of (n − 1)-cells in Y.

It remains to study the conjugacy classes of finite subgroups of groups of the form H_{L}:Γ and G_{L}:Γ. In fact it is no more difficult to study conjugacy classes of subgroups Q′ such that Q′ ∩G_{L} = {1}. Consider first the collection of subgroups Γ′ of G_{L}:Γ which map isomorphically to G_{L}:Γ/G_{L} ≅ Γ. The action of Γ on X_{L}/G_{L} fixes the unique vertex. It follows that each such Γ′ fixes some vertex v of X_{L}. Since the vertices form a single orbit, it follows that all such Γ′ are conjugate in G_{L}:Γ.

Proposition 8 Let Γ act on L, let Q ≤ Γ, and let Q′ be any subgroup of G_{L}:Γ that maps isomorphically to Q ≤ Γ = G_{L}:Γ/G_{L}. If L^{Q} = ∅, then Q′ fixes a unique vertex in X_{L}. If L^{Q} contains the barycentre of an m-simplex, and Q′ fixes a vertex (g) ∈ X_{L} of height f(g) = a, then Q′ also fixes a vertex of height a + (m + 1)n for each integer n.

Remark 9 Since we are not assuming that the action of Γ on L makes L into a Γ-CW-complex, it is not necessarily the case that L^{Q} is a subcomplex of L. However there can be a point of L^{Q} in the interior of the simplex σ only if qσ = σ for all q ∈ Q. In this case the barycentre of σ is a point fixed by Q.

Proof. For the first time, we shall make use of the CAT(0) metric on X_{L}. Suppose that Q′ fixes two distinct vertices (g), (h) of X_{L}. Since geodesics in a CAT(0) metric space are unique, it follows that the geodesic arc from (g) to (h) is also fixed by Q′. The start of this arc is a straight line passing from (g) into the interior of C, an n-cube of X_{L} for some n > 0, which must be preserved (setwise) by Q′. If C = (g′, v_{1}, …, v_{n}), then it follows that the (n − 1)-simplex (v_{1}, …, v_{n}) in L is (setwise) preserved by Q, and hence that L^{Q} ≠ ∅.

For the second statement, suppose that (g) is fixed by Q′, and that the m-simplex (v_{0}, …, v_{m}) in L is setwise fixed by Q. Then the long diagonal from (g) to (gv_{0}v_{1} ⋯ v_{m}) in the (m + 1)-cube (g, v_{0}, …, v_{m}) is an arc fixed by Q′, which connects two vertices whose heights differ by m + 1. It follows that for any n, the vertex g(v_{0}v_{1} ⋯ v_{m})^{n} is fixed by Q′.

Theorem 10 Let Γ act on L, and let Q ≤ Γ. If L^{Q} = ∅, then there are infinitely many conjugacy classes of subgroups Q′ of H_{L}:Γ whose members map isomorphically to conjugates of Q in Γ. If L^{Q} contains the barycentre of an m-simplex, then there are at most m + 1 conjugacy classes of such Q′ in H_{L}:Γ. In particular, if L^{Q} contains a vertex of L, then any two such subgroups are conjugate.

Proof. We know that any such Q′ fixes a vertex of X_{L} and that every vertex is fixed by some such Q′. In the case when L^{Q} = ∅, each Q′ fixes exactly one vertex of X_{L}. Since vertices of different heights are in different orbits for the action of H_{L}:Γ, it follows that in this case there are infinitely many conjugacy classes of such Q′.

In general, H_{L} acts transitively on the vertices of fixed height. If L^{Q} contains the barycentre of an m-simplex, and Q′ fixes a vertex of height a, then Q′ also fixes a vertex of height a + (m + 1)n for each n. Hence given any (m + 2) subgroups of H_{L}:Γ which map isomorphically to Q or one of its conjugates, some pair Q′, Q′′ of these subgroups must fix vertices of the same height. Let Γ′ ≥ Q′ and Γ′′ ≥ Q′′ be the stabilizers of these vertices, which map isomorphically to Γ. The subgroups Γ′ and Γ′′ are conjugate by an element of H_{L}. Hence it follows that Q′ and Q′′ are conjugate by some element of H_{L}:Γ.

3 Group actions

Here we construct the actions of finite groups Q and direct products of the form ℤ×Q on finite-dimensional simplicial complex that are needed in order to apply the constructions of the previous section. The first two propositions are included to show why actions of finite groups alone cannot give all the examples that we need.

Proposition 11 Suppose that Q is a finite group with normal subgroups P ≤ P′, so that P and Q/P′ are groups of prime power order and P′/P is cyclic. For any action of Q on a finite contractible complex L, the fixed point set L^{Q} is non-empty.

Proof. Let p and q be the primes (not necessarily distinct) so that |P | is a power of p and |Q : P | is a power of q. Let C denote P′/P, and let Q′ denote Q/P′.

By P. A. Smith theory [4, VII.10.5], the fixed point set L′ = L^{P} has the same mod-p homology as a point, and hence has Euler characteristic equal to 1. By character theory, it follows that the Euler characteristic of L′′ = L^{P′} = L′^{C} is equal to 1. By counting lengths of orbits of cells, one sees that L^{Q} = L′′^{Q′} has Euler characteristic congruent to 1 modulo q. This implies that L^{Q} is not empty.

The above proof also gives:

Proposition 12 Let Q be a finite group with a normal cyclic subgroup P′ so that Q/P′ is a group of prime power order. For any action of Q on a finite complex L′ with Euler characteristic χ(L′) = 1, the fixed point set L′^{Q} is non-empty.

The actions on ℚ-acyclic spaces that we shall need will all come from Theorem 14. In the proof of we shall need Lemma 13 concerning Wall’s finiteness obstruction.

Suppose that G is a group of type F P over the ring R, and that

0 → P_{n} → ⋯ → P_{1} → P_{0} → R → 0

is a resolution of R over RG by finitely generated projectives. As usual, let K_{0}(RG) denote the Grothendieck group of finitely-generated projective RG-modules. The Wall obstruction or Euler characteristic of G over R is the element of K_{0}(RG) given by the alternating sum

w(R, G) = ∑_{i}(−1)^{i}[P_{i}]

and is independent of the choice of resolution [10, I.7].

Lemma 13 Let Q be a finite group. The group G = ℤ × Q is F P over ℚ, and the Wall obstruction for this group is zero.

Proof. Let the group G = ℤ × Q act on the real line via the projection G → ℤ. There is a G-equivariant triangulation of the line with one orbit of 0-cells of type G/Q and one orbit of 1-cells, also of type G/Q. The cellular chain complex for this space gives a projective resolution for ℚ over ℚG of length one:

0 → ℚG/Q → ℚG/Q → ℚ → 0,

in which the modules in degrees 0 and 1 are isomorphic to each other.

Theorem 14 Let Q be a finite group, and let F be a non-empty family of subgroups of Q which is closed under conjugation and inclusion. There is a 3-dimensional ℚ-acyclic simplicial complex L admitting a cocompact action of Γ = ℤ × Q so that all cell stabilizers are finite and so that P ≤ Q fixes some point of L if and only if P ∈ F.

Proof. Let Δ be a finite set with a Q-action, such that Δ^{P} ≠ ∅ if and only if P ∈ F, and let Z = Q ∗ Q ∗ Δ be the join of two copies of Q and one copy of Δ, with the diagonal action of Q. This Z is a 2-dimensional simplyconnected Q-space, with the property that the Q-action is free except on the 0-skeleton. Let ℤ act on ℝ in the usual way, and let L_{0} be the product ℝ × Z with the product action of Γ = ℤ × Q. Now let L_{1} be the 2-skeleton of L_{0}. The cells of L_{1} in non-free orbits form a copy of ℝ × Δ with the product action of Γ. Let C_{∗} be the rational chain complex for L_{1}. Since L_{1} is 1-connected, C_{∗} forms the start of a projective resolution for ℚ over ℚΓ. As ℚΓ-modules, C_{2} is free and each of C_{1} and C_{0} is the direct sum of a free module and a copy of ℚ[ℤ × Δ]. Hence the element of K_{0}(ℚΓ) defined by the alternating sum [C_{2}]−[C_{1}]+[C_{0}] is in the subgroup of K_{0}(ℚΓ) generated by the free module. Since we know by Lemma 13 that the Wall obstruction for Γ over ℚ is zero, it follows that H_{2}(C_{∗}) is a stably-free ℚΓ-module. Make L_{2} by attaching finitely many free Γ-orbits of 2-spheres to L_{1} in such a way that H_{2}(L_{2}; ℚ) is a free ℚΓ-module. Let c_{1}, …, c_{k} be cycles in C_{2}(L_{2}, ℚ) representing a ℚΓ-basis for H_{2}(L_{2}; ℚ), and pick a large integer M so that each M.c_{i} is an integral cycle. Since L_{2} is 1-connected, each M.c_{i} is realized by the image of the fundamental class for S^{2} under some map f_{i} : S^{2} → L_{2}. Now define L_{3} by attaching free Γ-orbits of 3-balls to L_{2}, using the f_{i} as attaching maps for orbit representatives. This L_{3} has all of the required properties, except that it has been constructed as a Γ-CW-complex rather than as a Γ-simplicial complex. By the simplicial approximation theorem, we can construct a 3-dimensional Γ-simplicial complex L together with an equivariant homotopy equivalence L → L_{3}.

Before stating and proving Theorem 16, which will provide all the actions on contractible spaces that we shall need, we begin by establishing some notation, and proving a lemma concerning equivariant self-maps of spheres in linear representations. Lemma 15 and Theorem 16 were shown to the author by Bob Oliver.

Let S denote the unit sphere in ℂ^{n}, so that S is a sphere of odd dimension. For x ∈ S, let T_{x}S be the tangent space to S at x, and let B_{x} be the closed unit ball in T_{x}S, with boundary ∂B_{x}. For ϵ > 0, let e_{ϵ,x} : B_{x} → S denote the scalar multiple of the exponential map such that the image of B_{x} is a ball of radius ϵ in S. In the case when ϵ = π, this map sends the whole of ∂B_{x} to the point −x. The cases of interest to us include the case ϵ = π and the case when ϵ is small. Suppose that a finite group P acts linearly on S, fixing the point x. This induces a P -action on T_{x}S, and the exponential map e_{ϵ,x} is P -equivariant in the sense that e_{ϵ,x}(gv) = ge_{ϵ,x}(v) for all v ∈ B_{x} and all g ∈ P.

Each of the self-maps of spheres that we shall construct will have the property that it is equal to the identity except on a number of small balls. For such a map f : S → S, its support, supp(f), is defined to be the closure of the set of points x ∈ S so that f(x) ≠ x. Given another such map f′ : S → S with supp(f) ∩ supp(f′) = ∅, the map f ∐ f′ is defined by

f∐f′(x) = \begin{cases}f(x) & x ∈ supp(f) \\ f′(x) & x ∈ supp(f′) \\ x & x ∉ supp(f) ∪ supp(f′). \end{cases}

Suppose that a group Q acts on S. For f : S → S a self-map of S and g ∈ Q, define g ∗ f(s) = g(f(g^{−1}(s))). The support of g ∗ f is equal to g.supp(f).

For x ∈ S, let r : (B_{x}, ∂B_{x}) → (B_{x}, ∂B_{x}) be any map of degree −1, for example a reflection in a hyperplane through 0 in B_{x}. Define ϕ̃_{x}, ψ̃_{x} : B_{x} → S by

ϕ̃_{x}(v) = \begin{cases}−e_{π,x}(2v) & |v| ≤ 1/2 \\ (|v| − 1/2)v & |v| ≥ 1/2 \end{cases}

ψ̃_{x}(v) = \begin{cases}−e_{π,x}(r(2v)) & |v| ≤ 1/2 \\ (|v| − 1/2)v & |v| ≥ 1/2 \end{cases}

and define self-maps ϕ_{ϵ,x} and ψ_{ϵ,x} to be the identity outside of the image of e_{ϵ,x} and equal to ϕ̃_{x} ∘ e_{ϵ,x}^{−1} and ψ̃_{x} ∘ e_{ϵ,x}^{−1} respectively on their supports. If f is a self-map of S of degree n whose support is disjoint from the ϵ-ball around x, then f ∐ ϕ_{ϵ,x} is a self-map of degree n + 1 and f ∐ ψ_{ϵ,x} is a self-map of degree n − 1.

Suppose that a finite group Q acts linearly on S, so that the distance between any two points of the orbit Q.x is greater than 2ϵ. If g ∈ Q, then g∗ϕ_{ϵ,x} and ϕ_{ϵ,g.x} are equal. In particular, if g is an element of Q_{x}, the stabilizer of the point x, then the equation g ∗ ϕ_{ϵ,x} = ϕ_{ϵ,x} holds. Since the definition of ψ involved an arbitrary choice of function r, there is no corresponding equivariance property for the ψ self-maps. However, the map g ∗ ψ_{ϵ,x} is a self-map whose support is the ϵ-ball in S centred at g.x, and if f is any selfmap of S whose support is disjoint from this ball, the coproduct f ∐ g ∗ ψ_{ϵ,x} is a self-map whose degree is one less than that of f.

For any x ∈ S, define

Q.ϕ_{ϵ,x} = ∐_{g∈Q/Q_{x}} g ∗ ϕ_{ϵ,x},

for any sufficiently small ϵ, where the sum runs over a transversal to Q_{x} in Q. For x in a free Q-orbit, define

Q.ψ_{ϵ,x} = ∐_{g∈Q} g ∗ ψ_{ϵ,x},

for small ϵ. Each of these maps is Q-equivariant.

Lemma 15 Let S be the unit sphere in a complex representation of the finite group Q, and suppose that S contains points in Q-orbits of coprime lengths. Then S admits a Q-equivariant self-map of degree zero.

Proof. Without loss of generality, we may suppose that Q acts faithfully on S. The action of the unit circle in ℂ on S commutes with the Q-action, and so whenever S contains a Q-orbit of a given length, S contains infinitely many Q-orbits of that length. Pick points x_{1}, …, x_{m} in distinct Q-orbits, such that the sum of the lengths of the orbits is congruent to −1 modulo |Q|, i.e., so that there exists n with

|Q|n = 1 + ∑_{i=1}^{m} |Q.x_{i}|.

Now pick y_{1}, …, y_{n} in distinct free Q-orbits. Choose ϵ sufficiently small that any two points in any of these orbits are separated by more than 2ϵ. The coproduct

f = ∐_{i=1}^{m} Q.ϕ_{ϵ,x_{i}}∐ ∐_{j=1}^{n} Q.ψ_{ϵ,y_{j}}

is the required degree zero map.

Theorem 16 Let Q be a finite group not of prime power order. Then there exists a finite-dimensional contractible simplicial complex L with a cocompact action of ℤ × Q such that all stabilizers are finite and such that L^{Q} = ∅. Furthermore, L may be chosen in such a way that L^{P} ≠ ∅ for P any proper subgroup of Q.

Proof. Let S be the unit sphere in the ‘reduced regular representation of Q’, i.e., the regular representation ℂQ minus the trivial representation. This S has the property that S^{Q} = ∅ but S^{P} ≠ ∅ for any proper subgroup P < Q. Since Q is not of prime power order, S satisfies the hypotheses of Lemma 15, and so there exists a Q-equivariant map f : S → S of degree zero.

Take a Q-equivariant triangulation of the space I × S, where Q acts trivially on the interval I. By the simplicial approximation theorem, there is an integer n ≥ 0 and a simplicial map f′ : {1} × S^{(n)} → {0} × S which is equivariantly homotopic to f. Now let M be the nth barycentric subdivision of I × S relative to {0} × S. This is a copy of I × S, with the original triangulation on the subspace {0} × S and the nth barycentric subdivision of this triangulation on {1} × S. Construct L from the direct product ℤ × M by identifying (m, 1, s) with (m + 1, 0, f′(s)) for each s ∈ S and m ∈ ℤ. This space L is a triangulation of the doubly infinite mapping telescope of the map f′ : S → S. The fact that f′ has degree zero implies that L is contractible.

One difference between Theorem 14 and Theorem 16 is that the the dimension of the space constructed in Theorem 16 varies with Q. The final results in this section show that this difference cannot be avoided.

Lemma 17 Let Q be the special linear group SL_{n}(𝔽_{p}) over the field of p elements. Let e_{1}, …, e_{n} be the standard basis for the vector space 𝔽_{p}^{n}. Define elements τ_{1}, …, τ_{n} ∈ Q by

τ_{i}(e_{j}) = \begin{cases}e_{j} & i ≠ j \\ e_{i} + e_{i+1} & i = j < n \\ e_{n} + e_{1} & i = j = n. \end{cases}

The elements τ_{1}, …, τ_{n} generate Q, and any proper subset of them generates a subgroup of order a power of p.

Proof. Let θ be the cyclic permutation of the n standard basis elements, so that θ(e_{i}) = e_{i+1} for i < n and θ(e_{n}) = θ_{1}. The action of θ on Q by conjugation induces a cyclic permutation of the τ_{i}.

The elements τ_{1}, …, τ_{n−1} generate the upper triangular matrices, which form a Sylow p-subgroup of Q. This group contains each of the elementary matrices E_{i,j} for i < j, defined by

E_{i,j}(e_{k}) = \begin{cases}e_{k} & k ≠ i \\ e_{k} + e_{j} & k = i. \end{cases}

Conjugation by powers of θ induces a transitive permutation of the size n−1 subsets of τ_{1}, …, τ_{n}. Hence one sees that each such set generates a Sylow p-subgroup of Q.

It is well-known that the elementary matrices E_{i,j} for all i ≠ j form a generating set for Q. Each elementary matrix may be expressed as the conjugate of an upper triangular elementary matrix by some power of θ. It follows that the subgroup generated by τ_{1}, …, τ_{n} contains all elementary matrices and so is equal to Q.

Theorem 18 As in the previous lemma, let Q = SL_{n}(𝔽_{p}). Suppose that L is contractible, or that L is mod-p acyclic, and that Q acts on L so that L^{Q} = ∅. Then the dimension of L is at least n − 1.

Proof. We may assume that L is finite-dimensional, or there is nothing to prove. Let L_{i} be the fixed point subspace for the action of τ_{i}. By P. A. Smith theory, the fixed point set for the action of a p-group on a finitedimensional mod-p acyclic space is itself mod-p acyclic. From Lemma 17 it follows that each intersection of at most n − 1 of the L_{i} is mod-p acyclic, and that the intersection L_{1} ∩ … ∩ L_{n} is empty. Let X be the union of the L_{i}. The Mayer-Vietoris spectral sequence for the covering of X by the L_{i} with mod-p coefficients is isomorphic to the spectral sequence for the covering of the boundary of an (n − 1)-simplex by its faces. It follows that the mod-p homology of X is isomorphic to the mod-p homology of an (n − 2)-sphere. Hence X cannot be a subspace of a mod-p acyclic space of dimension strictly less than n − 1.

Remark 19 For a discrete group G, the minimal dimension of any contractible simplicial complex admitting a G-action without a global fixed point is an interesting invariant of G. The above theorem shows that this invariant can take arbitrarily large finite values. When G is a finite group of prime power order, the invariant takes the value infinity. Peter Kropholler has asked whether there are any other finitely generated groups G for which the invariant takes the value infinity.

4 Examples

Here we combine the results of Sections 2 and 3 to construct groups with strong homological finiteness properties that contain infinitely many conjugacy classes of certain finite subgroups.

Theorem 20 Let Q be a finite group not of prime power order. There is a group H of type F and a group G = H:Q such that G contains infinitely many conjugacy classes of subgroup isomorphic to Q and finitely many conjugacy classes of other finite subgroups.

Proof. By Theorem 16, there is a contractible finite-dimensional simplicial complex L with a cocompact action of ℤ × Q such that all stabilizers are finite, L^{Q} = ∅ and L^{P} ≠ ∅ if P < Q. Take a flag triangulation of L, and consider the Bestvina-Brady group H_{L}. By Theorem 7, the semi-direct product H = H_{L}:ℤ is of type F. By Theorem 10 the group G = H_{L}:(ℤ × Q) contains infinitely many conjugacy classes of subgroups isomorphic to Q and finitely many conjugacy classes of other finite subgroups.

We can now prove Theorem 2 as stated in the introduction. We first give a lemma concerning free products.

Lemma 21 Let G = G_{1} ∗ ⋯ ∗ G_{n} be a free product of groups, and let H_{i} be a finite-index normal subgroup of G_{i}. There is a bijection between conjugacy classes of non-trivial finite subgroups of G and the disjoint union of the sets of conjugacy classes of non-trivial finite subgroups of the G_{i}. The kernel of the map from G to ∏_{i} G_{i}/H_{i} is isomorphic to the free product of finitely many copies of the H_{i} and a finitely-generated free group.

Proof. Take a classifying space BG_{i} for each G_{i}, take a star-shaped tree with n edges whose central vertex has valency n, and make a classifying space BG for G by attaching the given BG_{i} to the ith boundary vertex of the tree. Now consider the regular covering of this space BG corresponding to the kernel of the homomorphism G → ∏_{i} G/H_{i}. This is a finite covering. The subspace of this covering lying above each BG_{i} is a finite disjoint union of copies of BH_{i}, and the subspace lying above the tree is a finite disjoint union of trees. Hence the whole space, which is a classifying space for the kernel, consists of a finite number of copies of the BH_{i}’s, connected together by a finite number of trees. The fundamental group of such a space is the free product of finitely many copies of the H_{i} and a finitely-generated free group.

For the claimed result concerning conjugacy classes of finite subgroups, we consider the tree obtained from the given expression for G as a free product. One way to construct this tree is by considering the universal covering space of the model for BG given above. This consists of copies of the EG_{i}’s, connected together by trees. Now contract each copy of EG_{i} to a point. The resulting G-space is contractible (since replacing EG_{i} by a single point does not change its homotopy type) and is 1-dimensional. It is therefore a G-tree, with n + 1 orbits of vertices and n orbits of edges. Each edge orbit is free, one of the vertex orbits is free, and there is one vertex orbit of type G/G_{i} for each 1 ≤ i ≤ n. Whenever a finite group acts on a tree, it has a fixed point. (To see this, take the finite subtree spanning an orbit, and peel off orbits of ‘leaves’ until the remainder is fixed.) Since the stabilizer of each edge is trivial, it follows that each non-trivial finite subgroup of G must fix exactly one vertex of the tree. This implies that each non-trivial finite subgroup of G is conjugate to a subgroup of exactly one of the G_{i}, and that two finite subgroups of G_{i} are conjugate in G if and only if they were already conjugate in G_{i}.

Proof. (Theorem 2.) Let Q = {Q_{1}, …, Q_{n}} be a finite list of isomorphism types of finite groups not of prime power order. For each Q_{i}, let G_{i} = H_{i}:Q_{i} be a group as in Theorem 20. Let G be G_{1} ∗ ⋯ ∗ G_{n}, the free product of the G_{i}. By Lemma 21, the group G is of type V F, contains infinitely many conjugacy classes of subgroup isomorphic to each Q_{i}, and contains finitely many conjugacy classes of finite subgroups of all other isomorphism types.

Theorem 22 Let Q be a non-trivial finite group. There exists a group G = H:Q of type F P over ℚ, containing infinitely many conjugacy classes of subgroups isomorphic to Q and finitely many conjugacy classes of other finite subgroups. Furthermore, H is torsion-free, has rational cohomological dimension at most 4 and has integral cohomological dimension at most 5.

Proof. By Theorem 14 there is a 3-dimensional ℚ-acyclic simplicial complex L with a cocompact ℤ × Q-action such that all stabilizers are finite, L^{Q} = ∅ and L^{P} ≠ ∅ if P < Q. Take a flag triangulation of L, and consider the Bestvina-Brady group H_{L}. By Theorem 7, the semi-direct product H = H_{L}:ℤ is F P over ℚ. By Theorem 10 the group G = H_{L}:(ℤ × Q) contains infinitely many conjugacy classes of subgroups isomorphic to Q and finitely many conjugacy classes of other finite subgroups. The rational cohomological dimension of H_{L} is at most the dimension of the ℚ-acyclic space Y appearing in Section 2, which is equal to the dimension of L, and the integral cohomological dimension of H_{L} is at most the dimension of the space X_{L}, which is one more than the dimension of L. The cohomological dimension of H_{L}:ℤ over any ring is at most one more than the cohomological dimension of H_{L} over the same ring.

Proof. (Theorem 4.) For each Q_{i} ∈ Q, Theorem 22 gives a group G_{i} of type F P over ℚ containing infinitely many conjugacy classes of subgroup isomorphic to Q and only finitely many conjugacy classes of other finite subgroups. By Lemma 21, the free product G = G_{1} ∗ ⋯ ∗ G_{n} is F P over ℚ, contains infinitely many conjugacy classes of subgroups isomorphic to each Q_{i} ∈ Q, and contains finitely many conjugacy classes of all other finite subgroups.

Remark 23 One difference between Theorems 2 and 4 is that each of the groups constructed in Theorem 4 has virtual cohomological dimension at most 5, whereas the virtual cohomological dimensions of the groups constructed in Theorem 2 seem to depend on the list Q. We do not know whether this necessarily happens, but the following proposition may be relevant.

Proposition 24 Suppose that G contains infinitely many conjugacy classes of subgroup isomorphic to SL_{n}(𝔽_{p}), and that G acts cocompactly with finite stabilizers on a mod-p-acyclic simplicial complex X. Then X must have dimension at least n − 1.

Proof. There are only finitely many orbits in X, and hence only finitely many conjugacy classes of subgroup of G can fix some point of X. It follows that there is a subgroup isomorphic to SL_{n}(𝔽_{p}) that has no fixed point, and we may apply Theorem 18 to deduce the required result.

Remark 25 If G is virtually torsion-free and acts cocompactly with finite stablizers on a contractible simplicial complex X, then G is of type V F. It seems to be unknown whether every group of type V F admits such an action. It also seems to be unknown whether every group of type F L over a prime field F admits a free cocompact action on an F -acyclic simplicial complex X. If F is not assumed to be a prime field, then there are counterexamples. In [6] we exhibited a group which is F L over ℂ but which is not F L over ℝ. This group cannot admit a cocompact free action on any ℂ-acyclic simplicial complex X.

We conclude this section with a brief discussion of the Grothendieck group K_{0}(ℚG) of finitely generated projective modules for ℚG and its connection with conjugacy classes of elements of finite order in G. First, we recall the definition of the Hattori-Stallings trace [1].

For any ring R, let T (R) denote the quotient of R by the additive subgroup generated by commutators of the form rs − sr for r, s ∈ R. For a square matrix A with coefficients in R, the Hattori-Stallings trace tr(A) is the element of T (R) defined as the equivalence class containing the sum of the diagonal entries of A. As an element of T (R), this satisfies the usual trace condition tr(AB) = tr(BA) for any matrices A and B.

Now suppose that P is a finitely generated projective R module, and that P is isomorphic to a summand of R^{n}. Pick an idempotent n × n matrix e_{P} whose image is isomorphic to P. The Hattori-Stallings rank of P is defined to be tr(e_{P}). It may be shown that this is independent of the choice of n and e_{P}. The Hattori-Stallings rank defines a group homomorphism from K_{0}(R) to T (R).

Theorem 26 For any group G, there is a subgroup of K_{0}(ℚG) which is free abelian of rank equal to the number of conjugacy classes of finite cyclic subgroups of G.

Proof. For the group algebra ℚG, the group T (ℚG) is the ℚ-vector space with basis the conjugacy classes of elements of G. For any finite cyclic subgroup C ≤ G, define an element e_{C} ∈ ℚG by

e_{C} = \frac{1}{|C|} ∑_{g∈C} g.

The element e_{C} is an idempotent, and the ℚG-module P_{C} defined by P_{C} = ℚGe_{C} is a projective ℚG-module. With respect to the basis for T (ℚG) given by the conjugacy classes of elements of G, the non-zero coefficients in the Hattori-Stallings trace for e_{C} are those corresponding to elements of C. If C_{1}, …, C_{n} are pairwise non-conjugate finite cyclic subgroups of G, it follows that the Hattori-Stallings traces e_{C_{1}}, …, e_{C_{n}} are linearly independent. It follows that the projectives of the form P_{C} generate a subgroup of K_{0}(ℚG) which is free abelian of rank equal to the number of conjugacy classes of finite cyclic subgroups of G.

Corollary 27 There are groups G of type V F for which K_{0}(ℚG) is not finitely generated.

Proof. Apply Theorem 26 to the groups with infinitely many conjugacy classes of finite cyclic subgroups constructed in Theorem 20.

5 Other properties of the groups

Suppose that Q is a group of automorphisms of a finite flag complex L with n vertices. It is shown in [7] that in this case the group G_{L}:Q is isomorphic to a subgroup of the special linear group SL_{2n}(ℤ). We do not know whether the groups G_{L}:Γ, for infinite L, are linear. Residual finiteness however is easier to establish.

Lemma 28 Suppose that Γ is residually finite and that Γ acts cocompactly and with finite stabilizers on a flag complex L. There is a finite-index normal subgroup Γ′ such that for any Γ′′ ≤ Γ′, the quotient L′ = L/Γ′′ is a flag simplicial complex.

Proof. There are finitely many conjugacy classes of simplex stabilizer in L, and each simplex stabilizer is finite. It follows that there is a finite-index normal subgroup Γ_{1} of Γ that acts freely on L. Since L is locally finite, there are only finitely many Γ_{1}-orbits of paths of length 1, 2 and 3 in the 1-skeleton of L. Hence we may pick Γ_{2} of finite-index in Γ_{1}, such that no two points in the same Γ_{2}-orbit are joined by an edge path of length less than four. We claim that we may take Γ′ = Γ_{2}.

If Γ′′ is any subgroup of Γ_{2}, then there is no edge path of length less than four between any two vertices in the same Γ′′-orbit. In particular, there can be no loops in L/Γ′. Hence every simplex of L maps injectively to a subspace of L/Γ′′. There can be no double edges in L/Γ′′, since that would give rise to an edge path of length two between vertices in the same Γ′′-orbit. Thus the 1-skeleton of L/Γ′′ is a simplicial complex.

Now suppose that v̄_{0}, …, v̄_{n} are a mutually adjacent set of vertices of L/Γ′′, and let v_{0} be a lift of v̄_{0}. There exists a unique lift v_{i} of each v̄_{i} that is adjacent to v_{0}. For each i ≠ j, there exists a unique g ∈ Γ_{2} so that v_{i} is adjacent to gv_{j}. But if g ≠ e, then the path (v_{j}, v_{0}, v_{i}, gv_{j}) gives rise to a contradiction. Thus the v_{i} are all adjacent to each other, and so there is a simplex σ of L with vertex set v_{0}, …, v_{n}. It follows that the quotient L/Γ′′ contains a simplex σ̄ spanning each complete subgraph of its 1-skeleton. Suppose that σ̄′ is any simplex of L/Γ′′ spanning the same complete subgraph as σ̄. There is a unique lift σ′ of σ̄′ containing v_{0}. If σ′ ≠ σ, then there exists i and g ≠ e so that gv_{i} is a vertex of σ′. But then there is an edge path of length 2 from v_{i} to gv_{i}. Hence any finite full subgraph of the 1-skeleton of L/Γ′′ is spanned by a unique simplex, and so L/Γ′′ is a flag complex.

Theorem 29 Let Γ be residually finite and let Γ act cocompactly and with finite stabilizers on a flag complex L. Then the group G_{L}:Γ is also residually finite.

Proof. Let g be a non-identity element of G_{L}:Γ. Since (G_{L}:Γ)/G_{L} is isomorphic to Γ, it suffices to consider the case when g ∈ G_{L}. Let K be a finite full subcomplex of L (i.e., a subcomplex containing as many simplices as possible given its 0-skeleton) such that g is in the subgroup generated by the vertices of K, and let J be a finite full subcomplex of L containing K and every vertex adjacent to a vertex of K. Let Γ′ be a finite-index subgroup of Γ as in Lemma 28, and let Γ′′ be a finite-index normal subgroup of Γ contained in Γ′ such that any two vertices of J lie in distinct Γ′′-orbits. Now M = L/Γ′′ is a finite flag complex, and K maps to a full subcomplex of L/Γ′′.

The group Γ/Γ′′ acts on M, and g has non-trivial image under the homomorphism G_{L}:Γ → G_{M}:(Γ/Γ′′). Since this group is isomorphic to a subgroup of SL_{2n}(ℤ), where n is the number of vertices of M (see corollary 8 of [7]), it follows that there is a finite quotient of G_{L}:Γ in which the image of g is non-zero.

In the special case when Γ = ℤ (which is the main case used earlier in the paper), we shall show how to describe the group G_{L}:Γ as the fundamental group of a finite locally CAT(0) cube complex. First we present two lemmas concerning right-angled Artin groups.

Lemma 30 Let N be a full subcomplex of a flag complex M. The inclusion i : N → M induces a split injection G_{N} → G_{M}.

Proof. The quotient of G_{M} by the subgroup generated by the vertices of M − N is naturally isomorphic to G_{N}.

Lemma 31 Let the flag complex K be expressed as K = L∪M, where L and M are full subcomplexes with N = L ∩ M. Then the group homomorphisms induced by the inclusion of each subcomplex in K induce an isomorphism G_{L} ∗_{G_{N}} G_{M} → G_{K}.

Proof. Immediate from the presentations of the groups, given the result of Lemma 30.

Theorem 32 Let Γ be an infinite cyclic group generated by γ, let Γ act on the flag complex L, and let M be a ‘fundamental domain’ for Γ in the sense that L = ⋃_{i} γ^{i}M. Define subcomplexes N_{0} and N_{1} by

N_{0} = γ^{−1}M ∩ M, N_{1} = M ∩ γM.

Then G_{L}:Γ is isomorphic to the HNN-extension G_{M}∗_{G_{N_{0}}=G_{N_{1}}}. (In this HNNextension, the base group is G_{M}, and the stable letter conjugates the subgroup G_{N_{0}} to the subgroup G_{N_{1}} by the map induced by γ : N_{0} → N_{1}.)

Proof. Let t denote the stable letter in the HNN-extension, and consider the homomorphism ϕ from the HNN-extension to ℤ that sends t to 1 and sends each element of G_{M} to 0. The kernel of ϕ is an infinite free product with amalgamation:

⋯ ∗ G_{−2} ∗_{H_{−1}} G_{−1} ∗_{H_{0}} G_{0} ∗_{H_{1}} G_{1} ∗_{H_{2}} G_{2} ∗ ⋯,

where G_{i} denotes t^{i}G_{M}t^{−i}, and H_{i} denotes t^{i}G_{N_{0}}t^{−i}. If we define M_{i} = γ^{i}M and N_{i} = γ^{i}N_{0}, there is an isomorphism ψ_{i} : G_{i} → G_{M_{i}} defined as the composite

G_{i} ⟶^{c(t^{−i})}G_{0}⟶^{1}G_{M} ⟶^{c(γ^{i})}G_{M_{i}}

of conjugation by t^{−i} followed by the identification of G_{0} and G_{M}, followed by conjugation by γ^{i}. Each of ψ_{i} and ψ_{i−1} induces an isomorphism from H_{i} to H_{N_{i}}, and these two are the same isomorphism. The ψ_{i} therefore fit together to make an isomorphism from ker(ϕ), described as an infinite free product with amalgamation, to the following infinite free product with amalgamation:

⋯ ∗ G_{M_{−2}} ∗_{H_{N_{−1}}} G_{M_{−1}} ∗_{H_{N_{0}}} G_{M_{0}} ∗_{H_{N_{1}}} G_{M_{1}} ∗_{H_{N_{2}}} G_{M_{2}} ∗ · · · .

Furthermore, this isomorphism is equivariant for the ℤ-actions given by conjugation by powers of t and γ. By Lemma 31, the inclusions of the G_{M_{i}} in G_{L} induce a Γ-equivariant isomorphism between the second free product with amalgamation and G_{L}. Hence we obtain an isomorphism G_{M}∗_{G_{N_{0}}=G_{N_{1}}} → G_{L}:Γ as required.

Corollary 33 Under the hypotheses of Theorem 32, the group G_{L}:Γ is the fundamental group of a finite locally CAT(0) cube complex.

Proof. For any flag complex K, let Y_{K} denote the explicit model for the classifying space BG_{K} described in Section 2, so that Y_{K} = X_{K}/G_{K}. The naturality properties of this construction are such that Y_{N_{0}} and Y_{N_{1}} are subcomplexes of Y_{M}. We construct a model Z for B(G_{L}:Γ) from Y_{M} and Y_{N_{0}} ×I by identifying {0} × Y_{N_{0}} with Y_{N_{0}} ⊆ Y_{M} via the identity map and identifying {1} × Y_{N_{0}} with Y_{N_{1}} ⊆ Y_{M} via the action of γ which gives an isomorphism from N_{0} to N_{1}.

The space Z as above is a model for B(G_{L}:Γ). To see that Z has the structure of a locally CAT(0) cube complex, one may either quote a gluing lemma (such as in [3], ref?), or one may show that the link of the unique vertex in Z is a flag complex, which suffices by Gromov’s lemma ([3], ref?). For any flag complex K, the link of the unique vertex in Y_{K} is a flag complex S(K), which is a sort of ‘double’ of K: each vertex v of K corresponds to two vertices v′, v′′ of S(K), and a set of vertices of S(K) is the vertex set of an n-simplex in S(K) if and only if its image in the vertex set of K is the vertex set of an n-simplex. (For example, in the case when K is a 2-simplex, S(K) is the boundary of an octahedron.) The link of the vertex in Z is isomorphic to S(M) with a cone attached to each of the subspaces S(N_{0}) and S(N_{1}), and hence it is a flag complex.

Corollary 34 Each of the groups G_{L}:(ℤ × Q) constructed in Section 4 acts cocompactly with finite stabilizers on some CAT(0) cube complex. In particular, there is a model for the universal space for proper actions of G_{L}:(ℤ × Q) which has finitely many orbits of cells.

Proof. Take a finite ‘fundamental domain’, M′, for the action of ℤ on L (as in the statement of Theorem 32). In case M′ is not Q-invariant, replace M′ by M = ⋃_{q∈Q} qM′. For this choice of M, there is a base-point preserving cellular Q-action on Z, the model for B(G_{L}:ℤ) constructed in Corollary 33. This induces the required action of G_{L}:(ℤ × Q) on the universal cover of Z. Whenever a group H acts with finite stabilizers on a CAT(0) cube complex, that space is a model for the universal space for proper actions of H [7].

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Author’s address: 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