Uncountably many quasi-isometry classes of groups of type F P

Robert P. Kropholler  Ian J. Leary^{∗} Ignat Soroko^{†}

January 26, 2018

Abstract

In [8] one of the authors constructed uncountable families of groups of type F P and of n-dimensional Poincaré duality groups for each n ≥ 4. We show that the groups constructed in [8] comprise uncountably many quasi-isometry classes. We deduce that for each n ≥ 4 there are uncountably many quasi-isometry classes of acyclic n-manifolds admitting free cocompact properly discontinuous discrete group actions.

1 Introduction

Throughout this article, the phrase ‘continuously many’ will be used to describe sets having the cardinality of the real numbers. In [8] one of the authors exhibited continuously many isomorphism types of groups of type F P, extending the work of Bestvina and Brady [1], who constructed the first examples of groups of type F P that are not finitely presented. We extend these results still further, by showing that the groups constructed in [8] fall into continuously many quasi-isometry classes.

Bestvina-Brady associate a group BB_{L} to each finite flag complex in such a way that the homological properties of the group BB_{L} are controlled by those of the flag complex L. In particular, in the case when L is acyclic but not contractible, BB_{L} is type F P but not finitely presented. In [8], a group G_{L}(S) is associated to each connected finite flag complex L and each set S ⊆ ℤ in such a way that the homological properties of G_{L}(S) are controlled by those of L and its universal cover, L̃. In the case when L and L̃ are both acyclic, each G_{L}(S) is type F P. The construction of G_{L}(S) generalizes that of BB_{L}, and in particular G_{L}(ℤ) is BB_{L}. Our first main theorem is as follows.

∗Partially supported by a Research Fellowship from the Leverhulme Trust.

†Partially supported by NSF Grants DMS-1107452, 1107263, 1107367 and ‘RNMS:Geometric Structures and Representation Varieties’ (the GEAR Network)

0This work was started at MSRI, Berkeley (during the program Geometric Group Theory), where research is supported by the National Science Foundation under Grant No. DMS-1440140. It was completed at INI, Cambridge, during the programme Nonpositive curvature, group actions and cohomology, where research is supported by EPSRC grant EP/K032208/1. The authors thank both MSRI and INI for their support and hospitality.

Theorem 1. For each fixed finite connected flag complex L that is not simply-connected, there are continuously many quasi-isometry classes of groups G_{L}(S).

The invariant that we rely on to distinguish the groups G_{L}(S) is the invariant that was introduced by Bowditch [2] in his construction of continuously many quasi-isometry classes of 2-generator groups. (Grigorchuk’s construction of such a family, using growth rate to distinguish the groups [7], is no help to us because the groups G_{L}(S) all have exponential growth.)

To a graph Γ Bowditch associates a set H(Γ) of natural numbers, consisting of the lengths of loops in Γ that are taut in the sense that they are not consequences of shorter loops. He describes the relationship between H(Γ) and H(Γ′) in the case when Γ and Γ′ are quasi-isometric. When Γ is the Cayley graph associated to a group presentation satisfying the C′(1/6) small cancellation condition, the set H(Γ) is equal to the set of lengths of the relators of the presentation.

Our proof involves estimating the set H(Γ(S)), where Γ(S) is the Cayley graph associated to the natural generating set for G_{L}(S). The natural presentation for G_{L}(S) contains relators whose lengths are parametrized by the absolute values of the members of S, but it also contains many relators of length 3, and does not satisfy the C′(1/6) condition. To apply Bowditch’s technique we need a lower bound for the word lengths of elements in the kernel of the map G_{L}(S) → G_{L}(T) for S ⊆ T, in terms of T − S. The Cayley graph Γ(S) embeds naturally in a CAT(0) cubical complex. Our lower bound on word length uses this embedding and an easy lemma concerning maps between CAT(0) spaces. In the statement, the singular set for a map consists of all points at which it is not a local isometry.

Lemma 2. Let f : X → Y be a continuous map of CAT(0) metric spaces, and suppose that x ≠ x′ but f(x) = f(x′). Then the distance d_{X}(x, x′) is at least twice the distance from x to the singular set for f.

In the final section of this article, we study the behaviour of the Bowditch length spectrum of a Cayley graph under the operation of taking certain semi-direct products with infinitely generated right-angled Coxeter groups. Such semi-direct products arise when applying Davis’s trick to the groups G_{L}(S) [5, 6]. For each S ⊆ ℤ one constructs a group J(S) which is the semi-direct product of G_{L}(S) and a suitable Coxeter group, together with maps J(S) → J(T) for S ⊆ T. Our aim is to estimate H(Λ(S)), where Λ(S) is a Cayley graph of the group J(S). Once again, the main problem is to give lower bounds for the word lengths of elements of the kernel of the map J(S) → J(T). Roughly speaking, we show that short elements in this kernel give rise to short elements in the kernel of G_{L}(S) → G_{L}(T). The main result of this section, Theorem 13, is too technical to state in this introduction, but instead we give one of its applications.

Corollary 3. For each d ≥ 4 there is a closed aspherical d-manifold admitting continuously many quasi-isometry classes of regular acyclic covers.

In contrast to the geometric methods used throughout the rest of the article, the proof of Theorem 13 is purely algebraic. It would be interesting to have a geometric proof of this theorem and conversely to have algebraic proofs of our results concerning G_{L}(S).

Acknowledgements

The authors thank Noel Brady and Martin Bridson for their interest in this project and for stimulating discussions about it. We also thank them both for having posed the question of whether the groups G_{L}(S) fall into continuously many quasi-isometry classes, which initiated the project.

2 Background

There are various types of Cayley graphs, but we shall need just one type which we shall call the simplicial Cayley graph Γ(G, S) associated to the group G and generating set S. This is the simplicial graph with vertex set G and edge set the 2-element sets of the form {g, gs} for some s ∈ S. This definition could be made for any S ⊆ G; the fact that S generates G is equivalent to the graph Γ(G, S) being connected. Any simplicial graph with a free, transitive G-action on its vertex set is isomorphic as a graph with G-action to Γ(G, S) for some S. The action of G on the edges of Γ(G, S) is free if and only if S contains no element of order two.

Next we recall some material from [2] concerning quasi-isometries and Bowditch’s taut loop length spectrum. Since we change the language slightly we restate some of the results from [2] in our terms: the proofs are identical to the originals. Each graph Γ that we consider will be connected, simplicial, and will be viewed as a metric space via the path metric d_{Γ}, in which each edge has length one. The induced metric on the vertex set of a Cayley graph Γ(G, S) is thus the S-word length metric on the group G. For k > 0 an integer, recall that a function f : X → Y between metric spaces is k-Lipschitz if d_{Y} (f(x), f(x′)) ≤ k.d_{X}(x, x′) for all x, x′ ∈ X. Following Bowditch [2], we say that graphs Γ and Λ are k-quasi-isometric if there exist a pair of k-Lipschitz maps of vertex sets ϕ : V (Γ) → V (Λ) and ψ : V (Λ) → V (Γ) so that d_{Γ}(x, ψ ∘ ϕ(x)) ≤ k for each vertex x of Γ and similarly d_{Λ}(y, ϕ ∘ ψ(y)) ≤ k for each vertex y of Λ. Graphs are quasi-isometric if they are k-quasi-isometric for some integer k > 0.

We remark that the above definition is not the standard one; see for example [3, I.8.14] for the standard definition of a quasi-isometry between metric spaces. We leave it as an exercise to check that graphs Γ, Λ are quasi-isometric as above if and only if the metric spaces (Γ, d_{Γ}) and (Λ, d_{Λ}) are quasi-isometric in the usual sense.

An edge loop of length l in a (simplicial) graph Γ is a sequence v_{0}, …, v_{l} of vertices such that v_{0} = v_{l} and {v_{i−1}, v_{i}} is an edge for 1 ≤ i ≤ l. For a graph Γ and an integer constant l, let Γ_{l} denote the 2-complex whose 1-skeleton is the geometric realization of Γ, with one 2-cell attached to each edge loop in Γ of length strictly less than l. An edge loop of length l in Γ is said to be taut if it is not null-homotopic in Γ_{l}. Bowditch’s taut loop length spectrum H(Γ) for the graph Γ is the set of lengths of taut loops.

We are interested in the 2-complex Γ_{l} only to define taut loops: if Γ′ is any subcomplex with Γ ⊆ Γ′ ⊆ Γ_{l} so that the induced map on fundamental groups π_{1}(Γ′) → π_{1}(Γ_{l}) is an isomorphism, then an edge loop is taut if and only if it is not null-homotopic in Γ′.

Bowditch defines subsets H, H′ ⊆ ℕ to be k-related if for all l ≥ k^{2} +2k +2, whenever l ∈ H then there is some l′ ∈ H′ with l/k ≤ l′ ≤ lk and vice-versa. He then proves

Lemma 4. If (connected) graphs Γ and Λ are k-quasi-isometric, then H(Γ) and H(Λ) are k-related.

In our terms, the lemmas that Bowditch uses to prove the above result are as follows.

Lemma 5. H(Γ) is equal to the set of l ∈ ℕ for which the induced map of fundamental groups π_{1}(Γ_{l}) → π_{1}(Γ_{l+1}) is not an isomorphism.

For any fixed l, let i_{Γ,l} denote the inclusion of Γ in the 2-complex Γ_{l}.

Lemma 6. If ϕ : Γ → Λ and ψ : Λ → Γ form a k-quasi-isometry between Γ and Λ, then for any l ≥ k^{2} + 2k + 2 there are homotopies

i_{Γ,l} ∘ ψ ∘ ϕ ≃ i_{Γ,l} and i_{Λ,l} ∘ ϕ ∘ ψ ≃ i_{Λ,l}.

Next we review some material from [1, 8]. A flag complex or clique complex is a simplicial complex in which every finite set of mutually adjacent vertices spans a simplex. Let 𝕋 denote the circle ℝ/ℤ, viewed as a CW-complex with one vertex at 0 + ℤ ∈ ℝ/ℤ and one edge. For a finite set V let 𝕋^{V} denote the product ∏_{v∈V} 𝕋_{v}, where 𝕋_{v} denotes a copy of 𝕋. Non-empty subcomplexes of 𝕋^{V} are in bijective correspondence with simplicial complexes with vertex set contained in V. If L is a flag complex with vertex set V, let 𝕋_{L} denote the corresponding subcomplex of 𝕋^{V}. This complex is aspherical, and its fundamental group is the right-angled Artin group A_{L} associated to L, with generators corresponding to the vertices of L, subject only to the relations that v and w commute whenever {v, w} is an edge of L.

The universal covering space X_{L} of 𝕋_{L} has a natural cubical structure, and is a CAT(0) cubical complex. The additive group structure in 𝕋 = ℝ/ℤ defines a map l : 𝕋^{V} → 𝕋, and hence a map l_{L} : 𝕋_{L} → 𝕋. Define 𝕋̃_{L} to be the regular covering of 𝕋_{L} induced by pulling back the universal covering of 𝕋 along l_{L}. The Bestvina-Brady group BB_{L} is defined to be the fundamental group π_{1}(𝕋̃_{L}), or equivalently the kernel of the map A_{L} → ℤ of fundamental groups induced by l : 𝕋_{L} → 𝕋. Bestvina and Brady showed that many properties of BB_{L} are determined by properties of L. In the case when L is acyclic but not simply connected, BB_{L} is type F P but not finitely presented [1].

Let f_{L} : X_{L} → ℝ be the map of universal coverings induced by l_{L}. This map has the following properties: if we identify each n-cube of X_{L} with [0, 1]^{n} then its restriction to each n-cube is equal to an affine map; the image of each vertex of X_{L} is an integer; the image of each n-cube of X_{L} is an interval of length n. We view f_{L} as defining a height function on X_{L}. There is a regular covering map X_{L} → 𝕋̃_{L}, with covering group BB_{L}.

In [8] this is generalized, under the extra assumption that L be connected. For each set S ⊆ ℤ, a CAT(0) cubical complex X_{L}^{(S)} is defined, together with a regular branched covering map X_{L}^{(S)} → 𝕋̃_{L}, and the group G_{L}(S) is by definition the covering group for this covering. The only branch points of this covering are the vertices of X_{L}^{(S)} whose height is not in S, and the stabilizer in G_{L}(S) of each branch point is a subgroup isomorphic to the fundamental group π_{1}(L). (In particular, the construction is non-trivial only when L is not simply-connected.) If S ⊆ T ⊆ ℤ, there is a regular branched covering map X_{L}^{(S)} → X_{L}^{(T)}, branched only at vertices of height in T − S, and the branched covering X_{L}^{(S)} → 𝕋_{L} factors through this. If S ⊆ T then there is a surjective group homomorphism G_{L}(S) → G_{L}(T), and the branched covering map X_{L}^{(S)} → X_{L}^{(T)} is equivariant for this homomorphism. The group G_{L}(ℤ) is equal to BB_{L}.

The height function on X_{L} induces a G_{L}(S)-invariant height function on X_{L}^{(S)} for each S ⊆ ℤ. Since 𝕋̃_{L} has only one vertex of each integer height, the group G_{L}(S) acts transitively on the vertices of X_{L}^{(S)} of each height. The intersection of the 2-skeleton of X_{L}^{(S)} and the 0-level set (i.e., the points of height 0) is a simplicial graph Γ whose 0-skeleton is the vertices of height 0. Orbits of edges in Γ correspond to A_{L}-orbits of squares in X_{L}, or equivalently to edges of L. If 0 ∈ S then G_{L}(S) acts freely on Γ, and so Γ can be identified with a simplicial Cayley graph for G_{L}(S). This gives a natural choice of generators for G_{L}(S) when 0 ∈ S, in bijective correspondence with the directed edges of L. Under the composite map G_{L}(S) → G_{L}(ℤ) = BB_{L} → A_{L} the element corresponding to the directed edge from vertex x to vertex y maps to the element xy^{−1}. To give a presentation for G_{L}(S) with this generating set, we first fix a finite collection Ω of directed loops in L that normally generates π_{1}(L). In other words, if one attaches discs to L along the loops in Ω, one obtains a simply-connected complex. Three families of relators occur in this presentation, which we call P (L, Ω):

• (Edge relations) for each directed edge a with opposite edge \overline{a}, the relation a\overline{a} = 1;

• (Triangle relations) for each directed triangle (a, b, c) in L the relations abc = 1 and a^{−1}b^{−1}c^{−1} = 1;

• (Long cycle relations) for each n ∈ S − {0} and each (a_{1}, …, a_{l}) ∈ Ω the relation a_{1}^{n}a_{2}^{n} ⋯ a_{l}^{n} = 1.

Another crucial property of these presentations is that only the long relations corresponding to n ∈ S hold in G_{L}(S): if (a_{1}, …, a_{l}) ∈ Ω is not null-homotopic in L, then a_{1}^{n}a_{2}^{n} ⋯ a_{l}^{n} ≠ 1 for n ∉ S ∪ {0}.

We close by giving some references for more general background material. For CAT(0) spaces we suggest [3], and for homological finiteness conditions such as the F P property we suggest [4]. Each of these topics is also covered briefly in the appendices to [6], which is our recommended source for Coxeter groups.

3 Bounding word lengths by CAT(0) distances

Our first task is to establish Lemma 2, as stated in the Introduction. Recall that a singular point of a map between CAT(0) spaces is a point at which the map is not a local isometry.

Proof. (Lemma 2). As in the statement, let x, x′ ∈ X be distinct points such that f(x) = f(x′), and suppose that the geodesic arc γ from x to x′ does not pass through the singular set. In this case, f ∘ γ is a locally geodesic arc in Y, whose end points are both equal to f(x). In a CAT(0) space any locally geodesic arc is a geodesic arc, and the unique geodesic arc from f(x) to f(x′) = f(x) is the constant arc of length 0. This contradiction shows that γ must pass through the singular set. The claim follows.

Lemma 7. Let L be a finite flag complex of dimension d. For any S ⊆ ℤ the distance from the 0-level set in the CAT(0) space X_{L}^{(S)} to a vertex of height n is |n|/\sqrt{d + 1}.

Proof. By symmetry it suffices to consider the case n > 0. Let γ be a path starting at a vertex v of height n, and moving at unit speed in X_{L}^{(S)} to the 0-level set. Minimizing the length of γ is equivalent to maximizing the speed of descent, i.e., minimizing the derivative of f_{L} ∘ γ.

The initial direction of travel of the path γ can be represented by a point of the link, Lk_{X}(v), of v in X = X_{L}^{(S)}. This is a simplicial complex in which each m-cube C of X_{L}^{(S)} that is incident on v contributes one (m − 1)-simplex, consisting of the unit tangent vectors at v that point into C.

If we identify an m-cube C of X_{L}^{(S)} with [0, 1]^{m}, then f_{L} restricted to C is equal to (t_{1}, …, t_{m}) ↦ t_{1} + t_{2} + ⋯ + t_{m} + r for some integer r. The gradient of f_{L} on the cube C is the vector (1, 1, …, 1), of length \sqrt{m}. Thus any path γ of fastest descent leaves v travelling in the direction of the long diagonal of a cube C of maximal dimension whose highest vertex is v.

If \sqrt{m} < n then the path will reach the unique lowest vertex v′ of C before it reaches the 0-level set; at this vertex a new choice of cube C′ of maximal dimension with v′ as its highest vertex should be made.

If w is any vertex of X = X_{L}^{(S)} then the cubes of X that have w as their highest vertex correspond to a subcomplex of Lk_{X}(w) called the descending or ↓-link, Lk_{X}^{↓}(w). Each descending link Lk_{X}^{↓}(w) is isomorphic to either L or its universal covering space L̃, depending only on whether the height of w lies in S. In particular, each descending link has dimension equal to d, the dimension of L.

It follows that we can always find at least one unit speed path γ starting at v with constant rate of descent \sqrt{d + 1} and there is no path descending faster. Hence the distance from v to the 0-level set is n/\sqrt{d + 1} as claimed.

For S ⊆ ℤ, define m(S) := min{|n| : n ∈ S}.

Lemma 8. Suppose that L is d-dimensional and that 0 ∈ S ⊆ T ⊆ ℤ, and take the standard generating set for G_{L}(S) and G_{L}(T). The word length of any non-identity element in the kernel of the map G_{L}(S) → G_{L}(T) is at least m(T − S)\sqrt{2/(d + 1)}.

Proof. The Cayley graph Γ(S) for G_{L}(S) is embedded in the 0-level set in X := X_{L}^{(S)}, and similarly Γ(T) is embedded in the 0-level set in Y := X_{L}^{(T)}. Moreover the branched covering map X → Y induces the natural quotient map Γ(S) → Γ(T). Let v be a height 0 vertex of X = X_{L}^{(S)}. Each standard generator for G_{L}(S) is represented by the diagonal of a square of X so for any g ∈ G_{L}(S) the triangle inequality tells us that the word length l(g) satisfies d_{X}(v, gv) ≤ \sqrt{2}l(g). Now g is in the kernel of the map to G_{L}(T) if and only if gv and v map to the same vertex of Y. Singular points for the map X → Y are vertices w whose heights lie in T − S, and by Lemma 7 these have d_{X}(v, w) ≥ m(T −S)/\sqrt{d + 1}. By Lemma 2 it follows that d_{X}(v, gv) ≥ 2m(T −S)/\sqrt{d + 1} and hence l(g) ≥ d_{X}(v, gv)/\sqrt{2} ≥ m(T − S)\sqrt{2/(d + 1)}.

4 Digression on convexity

The arguments used in the previous section can be used to show that the 0-level sets are very rarely convex or even quasi-convex. The material in this section is not needed for our main theorem.

Corollary 9. The 0-level set in X_{L}^{(S)} is convex if and only if L is a single simplex. In this case X_{L}^{(S)} = X_{L} does not depend on S.

Proof. If L is a d-simplex then X_{L}^{(S)} = X_{L} is a copy of ℝ^{d+1} and the 0-level set is an affine subspace. For the converse, if L is any flag complex other than a single simplex, then L will contain at least two maximal simplices. If v is any vertex of X_{L}^{(S)} of height one, the directions defined by the barycentres of these two maximal simplices give distinct geodesic paths from v to the 0-level set that are both locally distance minimizing, with end points x, x′ of height 0. Within the geodesic arc from x to x′, at most one point can locally minimize distance to v; the assumption that this geodesic arc lies in the 0-level set leads to a contradiction.

Corollary 10. If either L contains two simplices of maximal dimension d, or L̃ does and ℤ − S is infinite, then the 0-level set in X_{L}^{(S)} is not quasi-convex.

Proof. We give only a sketch. Let θ = θ(d) be the angle in ℝ^{d+1} between the vector (1, 1, …, 1) and one of the coordinate hyperplanes. Let v be a vertex of height N in X := X_{L}^{(S)}. If L has a unique simplex of dimension d, then N should be chosen in ℤ − S, otherwise N may be arbitrary. In this case there are two distance-minimizing geodesic paths from v to the 0-level set, with end points x and x′ as above, corresponding to leaving v in the directions given by the barycentres of two distinct d-dimensional simplices of the descending link. We view x, x′ and other points that depend on them as functions of N. The angle at v between these two geodesics is at least the constant 2θ. The geodesic triangle with vertices x, x′ and v is isoceles with angle at least 2θ between the two equal sides. The length of the equal sides is N/\sqrt{d + 1}. If y is the midpoint of the geodesic arc from x to x′, it follows that d_{X}(v, y) ≤ N cos(θ)/\sqrt{d + 1}. By increasing N this distance can be made arbitrarily smaller than N/\sqrt{d + 1}, the distance from v to the 0-level set. Hence for any k, there is an N so that y is not in the k-neighbourhood of the 0-level set. Thus the 0-level set is not quasi-convex.

5 Taut loop length spectra for G_{L}(S)

Throughout this section we fix a finite connected non-simply connected flag complex L. For S a subset of ℤ containing 0, let Γ(S) denote the Cayley graph of G_{L}(S) with respect to the standard generators. We give partial information concerning the taut loop length spectrum H(Γ(S)). As in [2], it will be convenient to assume that elements of S grow quickly.

Define α = α(L) by α = \sqrt{2/(d + 1)}, where d is the dimension of L. For a finite set Ω of loops in S that normally generates π_{1}(L), let β(L, Ω) be the maximum of the lengths of the loops in Ω, and define β(L) to be the minimum value of β(L, Ω) over all such Ω. Choose an integer constant C = C(L) so that C > β/α and Cα > 3. For F any subset of ℕ, define S(F) = {0} ∪ {C^{2^{n}} : n ∈ F }. With these definitions we prove a statement that is identical to [2, Proposition 1].

Proposition 11. If F, F′ are subsets of ℕ so that Γ(S(F)) and Γ(S(F′)) are quasiisometric then the symmetric difference of F and F′ is finite.

Theorem 1 follows immediately from this proposition. To prove the proposition we first describe H(Γ(S(F))).

Theorem 12. For any F ⊆ ℕ, the set H(Γ(S(F))) is contained in the disjoint union {3} ∪ ⋃_{n∈ℕ}[αC^{2^{n}}, βC^{2^{n}}]. The set H(Γ(S(F))) ∩ [αC^{2^{n}}, βC^{2^{n}}] is non-empty if and only if n ∈ F. Also 3 is in H(Γ(S(F))) if and only if L is not 1-dimensional.

Proof. The choice of C ensures that 3 < αC^{2^{0}} and that for all n βC^{2^{n}} < αC^{2^{n+1}} which implies that the union is disjoint. Since Γ = Γ(S(F)) is a simplicial Cayley graph, the edge relations a\overline{a} = 1 do not contribute to H(Γ), but the triangle relations imply that 3 ∈ H(Γ) whenever L has dimension at least two. If F = ∅ then the presentation P (L, Ω) contains only relations of length at most 3, so H(Γ) is either empty if L is 1-dimensional or is equal to {3} otherwise.

It remains to establish three statements

• If n ∈ F then H(Γ(S(F))) ∩ [αC^{2^{n}}, βC^{2^{n}}] ≠ ∅;

• If n ∉ F then H(Γ(S(F))) ∩ [αC^{2^{n}}, βC^{2^{n}}] = ∅;

• If k > 3 and k ∉ ⋃_{n∈ℕ}[αC^{2^{n}}, βC^{2^{n}}] then k ∉ H(Γ(S(F))).

The second and third of these statements can be grouped together into a single fourth statement:

• If k > 3 and k ∉ ⋃_{n∈F} [αC^{2^{n}}, βC^{2^{n}}] then k ∉ H(Γ(S(F))).

For the first statement, let F′ = F − {n}, and consider the covering map Γ(S(F′)) → Γ(S(F)). The group G_{L}(S(F′)) acts freely on Γ(S(F′)), so we may attach free orbits of 2-cells to Γ(S(F′)) to make a simply-connected Cayley 2-complex Δ. Now let K be the kernel of the map G_{L}(S(F′)) → G_{L}(S(F)), or equivalently the covering group for the regular covering Γ(S(F′)) → Γ(S(F)). The quotient Δ/K is a 2-complex with 1-skeleton the graph Γ(S(F)) and fundamental group K. We know that any non-identity element of K has word length at least αC^{2^{n}} and that there is a non-identity element of K of word length βC^{2^{n}}. The shortest non-identity element of K defines a loop in Γ(S(F)) ⊆ Δ/K that must be taut, since it is not null-homotopic in Δ/K whereas every strictly shorter loop in Γ(S(F)) is null-homotopic in Δ/K.

It remains to prove the fourth statement. Fix an integer k > 3 that is not an element of ⋃_{n∈F} [αC^{2^{n}}, βC^{2^{n}}]. Choose n ∈ ℕ maximal so that βC^{2^{n}} < k, if such n exists, and define n := −1 in the case when 3 < k < αC. Now let F′ := F ∩[0, n], where by definition [0, −1] = ∅. Once again, consider the covering map Γ(S(F′)) → Γ(S(F)). Since every relator in the presentation for G_{L}(S(F′)) has length at most βC^{2^{n}}, we may build a Cayley 2-complex Δ with 1-skeleton Γ(S(F′)) in which each 2-cell is attached to a loop of length at most βC^{2^{n}}. Now suppose that γ is a loop of length k in Γ(S(F)). If γ lifts to a loop in Γ(S(F′)) then it cannot be taut, since every loop in Γ(S(F′)) is null-homotopic in Δ. If on the other hand γ lifts to a non-closed path in Γ(S(F′)) then it corresponds to a non-identity element of the kernel of the map G_{L}(S(F′)) → G_{L}(S(F)) of word length at most k. But the shortest element in the kernel of this map has length at least αC^{2^{m}}, where m is the least element of S(F) − S(F′). By choice of n, we have that k ≤ βC^{2^{m}}, and by hypothesis k ∉ [αC^{2^{m}}, βC^{2^{m}}]. This contradiction shows that the loop γ cannot be taut.

Proof. (Propostion 11). For any l ∈ [αC^{2^{n}}, βC^{2^{n}}] and l′ ∈ [αC^{2^{n+m}}, βC^{2^{n+m}}] for some m > 0, we have that l′/l ≥ C^{2^{n}}. Hence if H(Γ(S(F))) and H(Γ(S(F′))) are k-related and n is in the symmetric difference of F and F′ then C^{2^{n}} < k.

6 Semidirect products

A right-angled Coxeter group is a group W admitting a presentation in which the only relators are that each generator has order two and that certain pairs of generators commute. A simplicial graph K gives rise to a right-angled Coxeter group W_{K}, with generators the set V (K) of vertices of K and as commuting pairs the ends of each element of the edge set E(K).

Now suppose that a group G acts as automorphisms of the graph K. This induces an action of G on W_{K} by automorphisms, permuting the given generators for W_{K}, and so there is a semidirect product group J = W_{K}⋊G. Identify G with its image inside the semidirect product J. A choice of generating set for G together with a choice of G-orbit representatives in V (K) gives rise to a generating set for J.

Now suppose that S ↦ G(S) is a functor from the category of subsets of ℤ with inclusions as morphisms to the category of finitely generated groups and surjective homomorphisms; for example S ↦ G_{L}(S) is such a functor for any connected finite flag complex L. Suppose further that G(∅) acts freely cocompactly on a (simplicial) graph K(∅) in such a way that any two vertices in the same G(∅)-orbit are at edge path distance at least four. For S ⊆ ℤ, define K(S) to be the quotient of K(∅) by the kernel of the map G(∅) → G(S), so that G(S) acts freely cocompactly on the graph K(S).

For S ⊆ ℤ, define J(S) to be the semidirect product W_{K(S)}⋊G(S). Then S ↦ J(S) is another functor from subsets of ℤ and inclusions to finitely generated groups and surjective group homomorphisms. Fix a finite generating set for J(∅) consisting of a finite generating set for G(∅) and a set of V′ of G(∅)-orbit representatives in V (K(∅)). As generating set for J(S), take the image of our given generating set for J(∅), and as generating set for G(S) take the image of our given generating set for G(∅). For each S, the generating set for G(S) is a subset of the generating set for J(S), and its complement consists of generators that are in the kernel of the map J(S) → G(S).

It will be useful to have a presentation for J(S) in terms of our generating set. Since G(∅) acts freely on the graph K(∅), the Coxeter relators between all of the generators for W_{K(∅)} are consequences of a finite set of relators indexed by the orbit representatives of vertices and edges in K(∅). To describe these relations, we choose a set E′ of G(∅)-orbit representatives of edges in K(∅), in such a way that each e ∈ E′ is incident on at least one v ∈ V′. For each u ∈ V = V (K(∅)), let g_{u} ∈ G(∅) be the unique element such that g_{u}.u ∈ V′. Now define an integer N_{1} by

N_{1} := max{l(g_{u}): u is incident on some edge in E′},

where l(g) denotes the word length of g ∈ G(∅). The relations in our presentation for J(S) are of the following kinds:

• v^{2} for each v ∈ V′;

• relators (vg_{u}ug_{u}^{−1})^{2}, where e ∈ E′, u and v are the vertices incident on e and v ∈ V′;

• the relators in a presentation for G(S).

Note that J(S) is finitely presented whenever G(S) is, and that the relators of the second kind are of length at most 4N_{1} + 4.

In the theorem below we write l_{J(S)} and l_{G(S)} for the word length with respect to these generating sets.

Theorem 13. Take notation and hypotheses as in the paragraphs above, and define N := 2N_{1}. For all S ⊆ T ⊆ ℤ, if w ∈ J(S) is in the kernel of J(S) → J(T) and l_{J(S)}(w) > N then there is g ∈ G(S) − {1} with l_{G(S)}(g) ≤ l_{J(S)}(w) so that g is in the kernel of G(S) → G(T).

Proof. In the case when w is not in the kernel of the map J(S) → G(S), we may take g = \overline{w}, the image of w, since this element is in the kernel of G(S) → G(T) and l_{G(S)}(g) ≤ l_{J(S)}(w).

Before starting the remaining (more difficult) case, we recall Tits’ solution to the word problem for a right-angled Coxeter group [6, Theorem 3.4.2]. If w = v_{1}v_{2} ⋯ v_{l} is a word in the standard generators for a right-angled Coxeter group that represents the identity, then w can be reduced to the trivial word using some sequence of moves of two types:

• if v, v′ are Coxeter generators that commute, replace the subword vv′ by v′v;

• replace a subword vv by the trivial subword.

The kernel of the map J(S) → G(S) is the right-angled Coxeter group W (S) := W_{K(S)}. Let w be in the kernel of this map as well as in the kernel of the map J(S) → J(T). Pick a shortest word in the generators for J(S) representing w, and write this word in the form w = h_{0}v_{1}h_{1}v_{2}h_{2} ⋯ h_{n−1}v_{n}h_{n}, where each v_{i} ∈ V′, each h_{i} ∈ G(S), and so that l_{J(S)}(w) = n + ∑_{i=0}^{n} l_{G(S)}(h_{i}). Now define g_{i} = h_{0}h_{1} ⋯ h_{i−1} for 1 ≤ i ≤ n. We have that

w = h_{0}v_{1}h_{1}v_{2}h_{2} ⋯ h_{n_{1}}v_{n}h_{n} = (g_{1}v_{1}g_{1}^{−1})(g_{2}v_{2}g_{2}^{−1}) ⋯ (g_{n}v_{n}g_{n}^{−1}).

This second expression for w will not be reduced in general, but each subword g_{i}v_{i}g_{i}^{−1} is equal to one of the standard Coxeter generators for the subgroup W (S). By hypothesis w is non-trivial in W (S) but is in the kernel of the map W (S) → W (T). Hence there must be a Tits move that can be applied to the image of this expression in W (T) that cannot be applied to the same expression in W (S).

If there is a Tits move of the second type that can be applied in W (T) but not in W (S), this implies that there exist i and j with 1 ≤ i < j ≤ n so that g_{i} = g_{j} ∈ G(T), but g_{i} ≠ g_{j} ∈ G(S). If on the other hand there is a Tits move of the first type that can be applied in W (T) but not in W (S), there exist 1 ≤ i < j ≤ n so that g_{i} = g_{j}g_{u} ∈ G(T) but g_{i} ≠ g_{j}g_{u} ∈ G(S), where g_{u} is one of the elements that takes a vertex of some edge in E′ to a vertex in V′, and so by definition of N_{1}, l_{G(S)}(g_{u}) ≤ l_{G(∅)}(g_{u}) ≤ N_{1}.

Define an element of J(S) by w′ := v_{i}h_{i}v_{i+1} ⋯ h_{j−1}v_{j}, where i and j are as above. Since the expression w = h_{0}v_{1}h_{1}v_{2}h_{2} ⋯ h_{n}v_{n} is of minimal length in terms of our generators for J(S), the length of the defining expression for w′ is also minimal. But l_{J(S)}(w′) is greater than or equal to the length of its image in G(S), h_{i}h_{i+1} ⋯ h_{j−1} = g_{i}^{−1}g_{j}. Thus l_{J(S)}(w′) ≥ l_{G(S)}(g_{i}^{−1}g_{j}). Depending which sort of Tits move was applied, g_{i}^{−1}g_{j} is either a non-trivial element of the kernel of the map G(S) → G(T) or differs from such an element by some g_{u}. Let g be this element of the kernel, and note that l_{G(S)}(g) ≤ l_{G(S)}(g_{i}^{−1}g_{j})+N_{1}. Since w maps to the identity element of G(S) and the image in G(S) of w′ is a path whose endpoints are at distance at least l_{G(S)}(g_{i}^{−1}g_{j}), we see that l_{J(S)}(w) ≥ 2l_{G(S)}(g_{i}^{−1}g_{j}). Putting these inequalities together we obtain l_{G(S)}(g) ≤ l_{J(S)}(w)/2 + N_{1}. If l_{J(S)}(w) > N = 2N_{1}, this implies that l_{G(S)}(g) ≤ l_{J(S)}(w) as required.

Now we specialize to the case of interest; the case when G(S) is the group G_{L}(S) for some finite flag complex L that is connected but not simply connected. In this case, Theorem 13 allows us to prove an analogue of Theorem 12 for the semidirect product J(S) := W_{K(S)}⋊G_{L}(S). We define constants α = α(L), β := β(L) and C := C(L) as in the previous section, and for F ⊆ ℕ we define S(F) ⊆ ℕ as before. Finally, we define M := 4N_{1} + 4, which depends on both L and on the action of G(∅) on K(∅), and we denote by Λ(S(F)) the Cayley graph Γ(J(S(F))).

Theorem 14. For any F ⊆ ℕ, the set H(Λ(S(F))) is contained in the union [0, M] ∪ ⋃_{n∈ℕ}[αC^{2^{n}}, βC^{2^{n}}]. If αC^{2^{n}} > M, then H(Λ(S(F))) ∩ [αC^{2^{n}}, βC^{2^{n}}] is non-empty if and only if n ∈ F.

Proof. Since the presentation for G_{L}(S(∅)) has only relators of length at most 3, we see that our presentation for J(S(∅)) consists of relators of length at most M. This verifies the claim in the case when F = ∅. As in the proof of Theorem 12, it suffices to verify two claims:

• If n ∈ F and αC^{2^{n}} > M then H(Λ(S(F))) ∩ [αC^{2^{n}}, βC^{2^{n}}] ≠ ∅;

• If k > M and k ∉ ⋃_{n∈F} [αC^{2^{n}}, βC^{2^{n}}], then k ∉ H(Λ(S(F))).

These statements can be verified exactly as in Theorem 12. For the first, we consider F′ := F − {n} and look at the covering map Λ(S(F′)) → Λ(S(F)). Attach free J(S)- orbits of 2-cells to Λ(S(F′)) to make a simply-connected Cayley complex Δ, and let K′ be the kernel of the map J(S(F′)) → J(S(F)). The quotient Δ/K′ has 1-skeleton Λ(S(F)) and fundamental group K′. As before, we know that there is an element of K′ ∩G(S(F′)) of length βC^{2^{n}} and that any non-identity element of this subgroup has length at least αC^{2^{n}}. Since M = 4N_{1} + 4 > N = 2N_{1}, Theorem 13 tells us that the word length of any non-identity element of K′ is also at least αC^{2^{n}}. Now the shortest non-identity element of K′ defines a taut loop in the required range.

For the second statement, given such a k, take n maximal so that βC^{2^{n}} < k, let F′ := F ∩ [0, n], and consider the covering map Λ(S(F′)) → Λ(S(F)). As before, we can build a Cayley 2-complex Δ with 1-skeleton Λ(S(F′)) in which each 2-cell is attached to a loop of length at most max{M, βC^{2^{n}}}. If γ is a loop in Λ(S(F)) of length k, then either γ is not taut, or the lift of γ to Λ(S(F′)) is a non-closed path. In the second case one obtains a non-identity element in the kernel of the map J(S(F′)) → J(S(F)) of length at most k. Since k > M > N, Theorem 13 tells us that there is a non-identity element of length at most k in the kernel of the map G(S(F′)) → G(S(F)), which cannot happen. This contradiction shows that γ cannot be taut.

The next two corollaries follow easily by the same proofs as in the previous section.

Corollary 15. If F, F′ are subsets of ℕ so that Λ(S(F)) and Λ(S(F′)) are quasiisometric, the symmetric difference of F and F′ is finite.

Corollary 16. For any L that is not simply-connected, and any graph K(∅) with a free G_{L}(∅)-action, there are continuously many quasi-isometry classes of groups J(S).

The reason why Corollary 16 is of value concerns the use of the Davis trick [6] to construct non-finitely presented Poincaré duality groups, as described in [8, Sec. 18]. The starting point is a 2-complex L for which each G_{L}(S) is type F P; for this group there is a finite 2-complex that is an Eilenberg-Mac Lane space K(G_{L}(∅), 1). For any n ≥ 4, one can find a compact n-manifold V with boundary that is also a K(G_{L}(∅), 1). Now let K be the 1-skeleton of the barycentric subdivision of a triangulation of the boundary of V, and let K(∅) be the 1-skeleton of the induced triangulation of the boundary of the universal cover of V, with G_{L}(∅) acting via deck transformations. For this choice of K(∅), the group J(∅) contains a finite-index torsion-free subgroup J′ that is the fundamental group of a closed aspherical n-manifold M, and such that regular covering M(S) of M with fundamental group the kernel of J′ → J(S) is acyclic for each S ⊆ ℤ. One deduces that each J(S) contains a finite-index torsion-free subgroup that is a Poincaré duality group of dimension n. Since the inclusion of a finite-index subgroup is always a quasi-isometry, Corollary 16 implies

Corollary 17. For each n ≥ 4 there are continuously many quasi-isometry classes of non-finitely presented n-dimensional Poincaré duality groups.

It remains to prove Corollary 3 from the introduction. This follows from the above discussion by the Schwarz-Milnor Lemma [3, I.8.19], which tells us that the acyclic covering manifold M(S) is quasi-isometric to J(S).

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