BRANCHED COVERINGS OF CUBICAL COMPLEXES AND

SUBGROUPS OF HYPERBOLIC GROUPS

Noel Brady

Abstract. By considering branched coverings of piecewise euclidean cubical complexes, we produce an example of a torsion free hyperbolic group containing a finitely presented subgroup which is not hyperbolic.

§1. Introduction

One of the fundamental notions in geometric group theory is that of a hyperbolic group. This concept was introduced in [Gr] as a generalization of both the fundamental group of a closed negatively curved manifold and of finitely generated free groups. Hyperbolic groups are finitely presented, and have solvable word and conjugacy problems. In addition, torsion free hyperbolic groups have finite Eilenberg-Mac Lane spaces, and so they are of finite type.

It is natural to inquire if hyperbolicity is an hereditary property of groups: i.e. to ask if subgroups of hyperbolic groups are themselves hyperbolic. The answer to this is clearly negative, as there are infinitely generated subgroups of the free group on two generators. For example, the kernel of the epimorphism F_{2} → ℤ which sends both generators of F_{2} to the generator 1 ∈ ℤ is not finitely generated.

E. Rips gave the first example of a hyperbolic group which has a finitely generated subgroup which is not finitely presentable, and hence not hyperbolic (see [R]). In [BM] there is an example of a finitely generated but not finitely presentable subgroup of a cocompact 4-dimensional Kleinian group. Now the question becomes: Is every finitely presented subgroup of a hyperbolic group necessarily hyperbolic?

In his fundamental paper [Gr], Gromov gave a sketch of what appeared to be a hyperbolic group G containing a finitely presented subgroup which is not hyperbolic. The group G is the fundamental group of a closed 5-dimensional manifold, realized as a finite branched cover of the 5-torus, branched over generically chosen codimension 2 subtori. The subgroup is the kernel of a homomorphism G → ℤ corresponding to a fibration of the 5-manifold over the circle. If the group G were hyperbolic, one could argue that the

Typeset by AMS-TEX

The author received partial support from NSF Grant DMS9200433 while this work was being carried out.

finitely presented subgroup was not hyperbolic. However, M. Bestvina [Be] has shown that the universal cover of any such branched cover of the 5-torus must contain isometrically embedded flat planes, and hence, that the group G is not hyperbolic.

There are situations where one is guaranteed that a finitely presented subgroup of a hyperbolic group is also hyperbolic. S. Gersten [Ge] showed that if a hyperbolic group has cohomological dimension 2 then every finitely presented subgroup is necessarily hyperbolic.

In this paper we produce a hyperbolic group which contains a finitely presented subgroup which is not hyperbolic. Here is the precise result.

Theorem 1.1. There exists a short exact sequence of groups

1 → H → G → ℤ → 1

such that,

(1) G is torsion free hyperbolic,

(2) H is finitely presented, and

(3) H is not of type FP_{3}.

In particular, H is not hyperbolic.

The group G in this example has cohomological dimension 3, and hence the bound in Gersten’s result is sharp.

The hyperbolic group G is obtained as the fundamental group of a compact, piecewise euclidean cubical complex Y, whose universal cover is a δ-hyperbolic metric space. The δhyperbolicity is ensured by taking a suitable branched covering of another cubical complex.

The epimorphism G → ℤ comes from a continuous map Y → S^{1}. This map lifts to the universal covers to give a “Morse function” f : X → ℝ. The kernel H acts cocompactly by deck transformations on point preimages or “level sets” of f. The Morse theory viewpoint developed in [BB] yields topological information about the level sets and hence finiteness properties of the kernel H.

This paper is organised as follows. In section 2 we define and describe properties of hyperbolic groups and of CAT(0) metric spaces. In Section 3 we define PE cubical complexes, and describe conditions on such complexes which ensure that their universal covers are CAT(0). We define the notion of a Morse function on a PE cubical complex in Section 4, and show how Morse functions can be used to deduce finiteness properties of certain groups. In Section 5 we define and study the notion of a branched covering of a PE cubical complex. In Section 6 branched coverings are used to construct a specific PE cubical complex together with a Morse function. The main result, Theorem 1.1, follows easily from this construction. Finally, in Section 7 we pose some interesting questions regarding finiteness properties of subgroups of hyperbolic groups.

We would like to thank Mladen Bestvina and Steve Gersten for the many interesting discussions we had during the course of this work.

§2. Hyperbolic Groups and CAT(0) Metric Spaces

Let G be a finitely generated group with finite generating set A ⊂ G which is closed with respect to inversion. We define the Cayley graph of G with respect to the generating set A (denoted by Γ_{A}(G)) as follows. The vertex set of Γ_{A}(G) is the group G, and for each a ∈ A and each g ∈ G there is a edge from the vertex g to the vertex ga. We make Γ_{A}(G) into a metric space by requiring that each edge be isometric to the unit interval, and then taking the induced path metric.

A path connected metric space X with distance function d is called a geodesic metric space if for all points x, y in X there is an isometric map from the interval [0, d(x, y)] to a path in X joining x to y. For example, if G is a finitely generated group then the Cayley graph of G with the usual path metric is a geodesic metric space. Like the example above, the geodesic metric spaces we consider will be complete and locally compact.

Given three points x, y, z in a geodesic metric space X, choose geodesics joining them to form a triangle Δxyz. Denote the geodesic sides of this triangle by [x, y], [x, z] and [y, z]. We say that the geodesic triangle Δxyz is δ-thin if the distance of any point on one side to the union of the other two sides is bounded above by δ. That is

d( p , [y, z] ∪ [x, z]) ≤ δ for all p ∈ [x, y]

and similarly for any permutation of the sides.

A path connected geodesic metric space X is called δ-hyperbolic if there exists (a universal) δ ≥ 0 such that every geodesic triangle in X is δ-thin. A finitely generated group G is called hyperbolic if the Cayley graph Γ_{A}(G) is a δ-hyperbolic metric space. In [S] it is shown that if B is another finite generating set then Γ_{B}(G) is δ′-hyperbolic for some δ′ ≥ 0, so the property of being a hyperbolic group is independent of the generating set.

We record some results about hyperbolic groups which will be used later on. The first result gives a useful way of recognizing a hyperbolic group. For a proof see [GdH] page 59.

Proposition 2.1. Suppose that a group G acts cocompactly, properly, by isometries on a δ-hyperbolic metric space. Then G is a hyperbolic group. □

The δ-hyperbolic spaces we consider in this paper are CAT(0) metric spaces which have no embedded flat planes. Let X be a metric space. Given three points x, y, z of X let x′, y′, and z′ be the vertices of a comparison triangle in the euclidean plane: that is, d(x, y) = d′(x′, y′), d(y, z) = d′(y′, z′), and d(z, x) = d′(z′, x′) where d( , ) denotes the metric on X and d′( , ) denotes the euclidean metric. We say that the complete, locally compact, geodesic metric space X satisfies the CAT(0) comparison inequaltiy or that X is a CAT(0) metric space if for all geodesic triangles xyz in X and points p ∈ [y, z] we have

d(x, p) ≤ d′(x′, p′)

where x′y′z′ is a comparison triangle for xyz and p′ ∈ [y′, z′] is such that d(p, y) = d′(p′, y′).

There is a nice way of characterizing those CAT(0) metric spaces which are also δhyperbolic in terms of embedded flat planes. See [Gr] and [Br].

Theorem 2.2 (Eberlein, Gromov, Bridson). Let X be a CAT(0) metric space which has a cocompact group of isometries, and which does not contain any isometric embeddings of the euclidean plane. Then X is a δ-hyperbolic metric space. □

The next result states that torsion free hyperbolic groups are of finite type. It will be used in Section 6 to conclude that the subgroup H in Theorem 1.1 is not hyperbolic. A proof can be found in [Gr] page 101 or [GdH] page 64.

Theorem 2.3 (Rips). Let H be a hyperbolic group. There exists a locally finite, contractible, finite dimensional simplicial complex (called the Rips complex of H) on which H acts faithfully, properly, simplicially, and cocompactly. In particular, if H is torsion free, then the action is free and the quotient of this complex by H is a finite Eilenberg-Mac Lane space K(H, 1). □

§3 Piecewise euclidean cubical complexes

Our construction uses piecewise euclidean complexes. We will need to know when the universal covers of such complexes become CAT(0) metric spaces in the induced path metric. It turns out that a simple combinatorial condition involving the notion of a flag complex suffices.

Definition 3.1. A simplicial complex L is said to be a flag complex if every finite collection of vertices of L which are pairwise adjacent spans a simplex in L. We see that such complexes L are completely determined by their 1-skeleton.

There are many examples of flag complexes. In fact, the first barycentric subdivision of any simplicial complex is a flag complex. We leave the verification of this as an exercise for the reader.

By a regular n-cube ⊔ ⊓^{n} we mean a cube which is isometric to the cube in ℝ^{n} with vertex at the origin, and edges defined by the unit basis vectors.

Definition 3.3. A piecewise euclidean cubical complex (of dimension ≤ N) is a cell complex X satisfying the following conditions.

• The characteristic map of an n-cell e (n ≤ N) is a map χ_{e} : ⊔ ⊓^{n} → e, where ⊔ ⊓^{n} is a regular n-cube in ℝ^{N}. A map f : ⊔ ⊓^{n} → e is said to be an admissible characteristic map if it is just χ precomposed with a partial isometry of ℝ^{N}.

• The restriction of χ_{e} to any face of ⊔ ⊓^{n} is an admissible characteristic map of a cell of X.

We shall refer to these as PE cubical complexes. Informally, we think of a finite PE cubical complex as being constructed from a finite, disjoint collection of regular cubes, by glueing their faces via isometries.

We now examine the local geometry of a PE cubical complex.

Definition 3.4. Let f be a face of a regular cube ⊔ ⊓^{n} ⊂ ℝ^{n}, and let x ∈ f be the center of this face. In the case f is a vertex, we just let x be this vertex. Define the link of f in ⊔ ⊓^{n} (denoted by Lk(f, ⊔ ⊓^{n})) to be the set of all unit tangent vectors at x ∈ ℝ^{n} which are orthogonal to f and which point into ⊔ ⊓^{n}. It is a subset of the unit sphere which is homeomorphic to a simplex of dimension n−(dim(f)+1). It has a natural spherical metric in which all dihedral angles are right angles. Such spherical simplices are termed all right simplices.

Thus the link of a cell in a PE cubical complex inherits the satructure of a piecewise spherical cell complex, all of whose cells are all right simplices. The next definition gives a combinatorial version of nonpositive curvature for PE cubical complexes.

Definition 3.5. A PE cubical complex is said to be nonpositively curved if the link of each vertex is a flag complex.

The following lemma of Gromov [Gr] (pg. 122) (see also [Da] Lemma 9.3) tells when a piecewise spherical cell complex, whose cells are all right simplices, satisfies the CAT(1) inequality. (Gromov’s statement uses the “no triangle condition” instead of the “flag complex” condition below).

Lemma 3.6 (Gromov). Let L be an all right, piecewise spherical complex. Then L is a CAT(1) space if and only if it is a flag complex. □

A PE cubical complex whose links are all CAT(1) spaces satisfies a local CAT(0) inequality (see [Gr] pg. 120 or [GH] pg. 197), and so its universal cover is a CAT(0) metric space (and hence is contractible) — see [Gr] pg. 119 or [GH] pg. 193. There is a nice survey of these results in section I of [Da].

§4. Morse Theory and Finiteness Properties of Groups

We will consider the following finiteness conditions for groups. A group H is said to be of type F_{n} if there is a K(H, 1) with finite n-skeleton. A group H is said to be of type FP_{n} if there exists a resolution (exact sequence)

P_{n} → P_{n−1} → ⋯ → P_{0} → ℤ

of the trivial ℤH-module ℤ by finitely generated projective ℤH-modules P_{i}.

An application of Schanuel’s Lemma from homological algebra gives us the following method of showing that a group H is of type FP_{n} but not of type FP_{n+1}. A proof of this can be found in K. Brown’s book [Br 1] (pp. 192-193) for example.

Proposition 4.1. If there exists a resolution

0 → Z_{n} → P_{n} → ⋯ → P_{0} → ℤ → 0

where the P_{i} are all finitely generated, projective over ℤH and Z_{n} is not finitely generated over ℤH. Then H is of type FP_{n} but not of type FP_{n+1}. □

The following standard result shows how one can use topological information about a group acting on a cell complex to conclude that the group satisfies various finiteness conditions.

Proposition 4.2. Suppose that the group H acts freely, properly, cellularly, and cocompactly on the cell complex X. Then the following implications hold.

(1) If H̃_{i}(X, ℤ) = 0 for 0 ≤ i ≤ n − 1 and H̃_{n}(X, ℤ) is not finitely generated as a ℤH-module, then H is of type FP_{n} but not of type FP_{n+1}.

(2) If X is (n − 1)-connected, then H is of type F_{n}. □

As an example, a torsion free hyperbolic group has a finite Eilenberg-Mac Lane space by Theorem 2.3 and so is of finite type. In particular, we see that it is of type FP_{n} for all integers n ≥ 0. This is how we show that the torsion free subgroup H in our example is not hyperbolic. If H were hyperbolic it would be of type FP_{3}, contradicting Theorem 1.1 (3).

As indicated in the introduction, we shall produce groups satisfying certain finiteness conditions by studying Morse functions on PE cubical complexes. A general version of Morse theory for affine complexes was introduced in [BB] where it was used to study finiteness properties of certain subgroups of right angled Artin groups. We recall the relevant definitions and results below for the category of PE cubical complexes.

Definition 4.3. Suppose that the group G acts freely, cocompactly, properly, by isometries on a contractible PE cubical complex X. Let ϕ : G → ℤ be an epimorphism, and let ℤ act on ℝ by integer translations.

We say that the continuous map f : X → ℝ is a ϕ-equivariant Morse function on X if it satisfies the following properties.

• f ∘ g = ϕ(g) ∘ f for all g ∈ G.

• For each n-cell e of X with characteristic map χ_{e} : ⊔ ⊓^{n} → X, the composition

fχ_{e} : ⊔ ⊓^{n} → ℝ

extends to an affine map ℝ^{n} → ℝ, and fχ_{e} is constant only for n = 0.

• The f-image of the 0-skeleton of X is discrete in ℝ.

We drop the adjective ϕ-equivariant if there are no group actions, and f still satisfies properties two and three.

It is observed in [BB] that PE cubical complexes have a natural PL structure. Thus, given a cell e in a PE complex X with characteristic map χ_{e} : ⊔ ⊓_{e} → e, and a vertex w ∈ ⊔ ⊓_{e} mapping to v ∈ e, we have a well-defined map of links

χ_{e∗} : Lk(w, ⊔ ⊓_{e}) → Lk(v, e) ⊂ Lk(v, X)

We can view Lk(v, X) as a union

Lk(v, X) = ⋃{ χ_{e∗}(Lk(w, ⊔ ⊓_{e})) | e is a cell of X and χ_{e}(w) = v }

The following definitions are by analogy with the notions of ascending and descending manifolds in Morse theory.

Definition 4.4. The ascending link (the ↑-link) is

Lk_{↑}(v, X) = ⋃{ χ_{e∗}(Lk(w, ⊔ ⊓_{e})) | χ_{e}(w) = v and fχ_{e} has a minimum at w } ⊂ Lk(v, X)

and the descending link (the ↓-link) is

Lk_{↓}(v, X) = ⋃{ χ_{e∗}(Lk(w, ⊔ ⊓_{e})) | χ_{e}(w) = v and fχ_{e} has a maximum at w } ⊂ Lk(v, X).

We will use the local link structure to deduce topological information about point preimages or level sets of Morse functions. First we fix notaion. Let f : X → ℝ be a Morse function as above, and J ⊂ ℝ a closed interval. Denote by X_{J} the preimage f^{−1}(J). When J consists of a single point t ∈ ℝ we denote the preimage by X_{t} and refer to it as a level set of f.

The following lemma and corollary are analogues of the basic fact that a Morse function on a manifold gives a handlebody decomposition of the manifold. These are stated and proven in [BB] as Lemma 2.5 and Corollary 2.6 respectively.

Lemma 4.5. Let f : X → ℝ be a Morse function on a PE cubical complex. Suppose J ⊂ J′ are closed and connected intervals in ℝ, inf J = inf J′, and J′ \J contains only one point r of f(0-cells). Then X_{J′} is homotopy equivalent to X_{J} with the copies of Lk_{↓}(v, X) (v a vertex with f(v) = r) coned off. □

A similar statement holds when inf J = inf J′ is replaced by sup J = sup J′ and Lk_{↓}(v, X) by Lk_{↑}(v, X).

Corollary 4.6. Let f : X → ℝ be a Morse function on a PE cubical complex. Suppose that J ⊂ J′ are intervals in ℝ.

(1) If each ↑-link and each ↓-link is homologically n-connected, then inclusion X_{J} ↪ X_{J′} induces isomorphism in H̃_{i} for i ≤ n and epimorphism in H̃_{n+1}.

(2) If each ↑-link and each ↓-link is simply connected, then inclusion X_{J} ↪ X_{J′} induces isomorphism in π_{1}. □

The theorem below demonstrates the power of the Morse theory point of view. Simple topological conditions on the links give rise to finiteness properties for the kernel subgroup. Most of this result appears as Theorem 4.1 of [BB].

Theorem 4.7. Let f : X → ℝ be a ϕ-equivariant Morse function and let H = Ker(ϕ) be as above.

(1) Suppose that the reduced homology of each ↑-link and each ↓-link is zero in all dimensions 0 through n + 1, except for dimension n. Then H is of type FP_{n} but is not of type FP_{n+1}.

(2) If all ↑-links and ↓-links are simply connected, then H is finitely presented.

Proof. The main point of this argument is that we have a contractible space X = f^{−1}((−∞, ∞)) expressed as the direct limit of spaces X_{[t−N,t+N]} for N ∈ ℕ and a given t ∈ ℝ.

Now, Corollary 4.6 (1) implies that H̃_{i}(X_{[t−N,t+N]}, ℤ) is isomorphic to H̃_{i}(X_{t}, ℤ) for 0 ≤ i ≤ n − 1, and all N ∈ ℕ. Since homology commutes with direct limits, this implies that H̃_{i}(X_{t}, ℤ) = 0 for 0 ≤ i ≤ n−1. We now show that H̃_{n}(X_{t}, ℤ) is not finitely generated as a ℤH-module. Part (1) then follows immediately from Proposition 4.2 (1).

Suppose to the contrary, that the ℤH-module H̃_{n}(X_{t}, ℤ) has a finite basis, with cycle representatives

z_{1}, …, z_{k} .

Then there exists an integer N ∈ ℕ such that each of the z_{i} bound (are null homologous) in X_{[t−N,t+N]}. Since H acts by “horizontal” translations on X, this implies that the inclusion induced map

H̃_{n}(X_{t}, ℤ) → H̃_{n}(X_{[t−N,t+N]}, ℤ)

is trivial. But by Corollary 4.6 (1) it is also an epimorphism. Thus H̃_{n}(X_{[t−N,t+N]}, ℤ) = 0. Let J ⊂ ℝ be a closed interval such that [t − N, t + N] ⊂ J and ℤ ∩ (J \ [t − N, t + N]) ≠ ∅. Lemma 4.5 implies that X_{J} can be obtained (upto homotopy) from X_{[t−N,t+N]} by coning off successive copies of the ascending and descending links of f. Let L denote such a link. The Mayer-Vietoris sequence together with the hypothesis H̃_{n}(L, ℤ) ≠ 0 implies that H̃_{n+1}(X_{J}, ℤ) ≠ 0 for all such intervals J. Now the hypothesis H̃_{n+1}(L, ℤ) = 0 together with the Mayer-Vietoris sequence shows that the inclusion induced maps

H̃_{n+1}(X_{J}, ℤ) → H̃_{n+1}(X_{J′}, ℤ)

are injective for all intervals J ⊂ J′ ⊂ ℝ. This implies that H̃_{n+1}(X, ℤ) ≠ 0 a contradiction.

Now for part (2). Since the links are 1-connected, they are connected. The argument in (1) above implies that the level sets X_{t} are also connected. Corollary 4.6 (2) implies that the inclusion X_{t} ↪ X induces an isomorphism in π_{1}. Hence X_{t} is 1-connected, and the finite presentability of H follows from Proposition 4.2 (2). □

§5. Branched Coverings of Piecewise Euclidean Cubical Complexes

We will construct the example in §6 by taking a “branched covering” of a particular PE cubical complex. In this section, we define and study the operation of taking a branched covering of a finite PE cubical complex.

We develop the notion of branched covering by analogy with the branched covering of a 3-manifold over a link. The first problem one notices is that a subcomplex of a PE cubical complex need not have a nice neighborhood like a link in a 3-manifold. We consider the following subcomplexes.

Definition 5.1. Let K be a PE cubical complex, and L ⊂ K a subcomplex. We say that L is a branching locus if it satisfies the following two conditions.

• For each m-cell χ_{e} : ⊔ ⊓_{e}^{m} → e of K with e ∩ L ≠ ∅ we have that χ_{e}^{−1}(L) is a disjoint union of faces of ⊔ ⊓_{e}^{m} (possibly all of ⊔ ⊓_{e}^{m}).

• Lk(e, K) \ Lk(e, L) is nonempty and connected for each cell e ⊂ L.

The first condition is a local convexity condition for L in K. It is used to describe the neighborhoods of L in K, and to prove that nonpositive curvature is preserved on passing to branched coverings in Lemma 5.5 below.

The second condition is a reformulation of the “codimension 2” condition in branched coverings of manifolds over submanifolds. It ensures that the trivial branched covering of a PE cubical complex K over a branching locus L is just K itself.

Definition 5.2. Let K be a PE cubical complex and L ⊂ K be a branching locus as above. By a finite branched covering of K over L we mean the result of the following process.

• Take a finite covering of K \ L.

• Lift the local piecewise euclidean metric on K \ L to this cover, and then consider the induced path metric.

• Finally, take the metric completion of this cover.

Let X be a metric space and Y ⊂ X. By an ϵ-neighborhood of Y in X—denoted by N_{ϵ}(Y, X)—we mean the set of all points of X which are within ϵ of Y.

For L ⊂ K as above and C a component of L, we see that the neighborhoods N_{ϵ}(C, K) are all disjoint for 0 < ϵ < 1/2. We can describe these neighborhoods as follows.

Suppose that C ⊂ L is a component of L, and χ_{e} : ⊔ ⊓_{e}^{m} → e is an m-cell of K such that ∅ ≠ χ_{e}^{−1}(C) is a proper face of ⊔ ⊓_{e}^{m}. Then, for 0 < ϵ < 1/2 there is a natural continuous map

f_{e} : ∂N_{ϵ}(χ_{e}^{−1}(C), ⊔ ⊓_{e}^{m}) → χ_{e}^{−1}(C)

defined by nearest point retraction. Note that N_{ϵ}(χ_{e}^{−1}(C), ⊔ ⊓_{e}^{m}) is homeomorphic to the mapping cylinder of f_{e}. The composites χ_{e}f_{e} paste together to define a continuous map

f_{C} : ∂N_{ϵ}(C, K) → C

and the neighborhood N_{ϵ}(C, K) is homeomorphic to the mapping cylinder M(f_{C}).

We say that the branched covering is trivial if the finite cover in step 2 of the definition above is a trivial cover. The following Lemma shows that the operation of finite branched coverings does not take one out of the class of finite PE cubical complexes.

Lemma 5.3. Let K be a finite PE cubical complex, and let K̂ be a finite branched covering. Then K̂ is also a finite PE cubical complex, and there is a natural continuous surjective map b : K̂ → K. In the case of a trivial branched covering, one has K̂ = K.

Proof. Note that K \ L is the union of a finite collection of “cubes” (that is regular cubes, possibly with some lower dimensional faces removed) glued together by isometries of their faces. Each of these “cubes” is 1-connected and so lifts to the finite cover \overline{K \ L} of K \ L. Thus, this finite cover is a finite union of such “cubes” glued together by isometries.

Now, the path metric on the finite covering is locally Euclidean. Thus, on forming the completion one just adds faces back to the “cubes” in \overline{K \ L}, and so K̂ inherits the structure of a finite PE cubical complex.

We shall show that the covering map \overline{K \ L} → K \ L extends to a continuous surjective map b : K̂ → K.

Since the covering map \overline{K \ L} → K \ L is locally distance nonincreasing, the induced map b : K̂ → K is well-defined and continuous.

Let x be a point in a cell e ⊂ L. Since Lk(e, K) \ Lk(e, L) is nonempty, there is a cell e_{1} ⊂ K such that e ⊂ e_{1} and χ_{e_{1}}^{−1}(L) is contained in the boundary of ⊔ ⊓_{e_{1}}. Any path in the interior of e_{1} which ends at x lifts to a path in the finite cover. Thus a Cauchy sequence in this path which converges to x lifts to a Cauchy sequence defining a point x̂ ∈ K̂ such that b(x̂) = x. Therefore b is surjective.

Finally, let s_{1} and s_{2} denote two Cauchy sequences of points in K \L, which are Cauchy equivalent in K and which define a point x ∈ L. Since ∅ ≠ (Lk(e, K) \ Lk(e, L)) is connected for each cell e ⊂ L, then s_{1} and s_{2} will still be Cauchy equivalent in the induced path metric of K \ L. Thus the completion of the trivial cover K \ L yields the complex K as required. □

If C is a component of the branching locus L and 0 < ϵ < 1/2. Then the deleted ϵ-neighborhood N_{ϵ}(C, K) \ C of C lifts to the finite cover to give deleted ϵ-neighborhoods of components of the preimage of C in K̂.

The next lemma derives a consequence of the local convexity condition in the definition of branching locus. It will be used in the proof of Lemma 5.5 below.

Lemma 5.4. Let K be a finite PE cubical complex of nonpositive curvature, L ⊂ K a branching locus, and v ∈ L a vertex. Then Lk(v, L) is a full subcomplex of Lk(v, K).

Proof. Since K is of nonpositive curvature the links Lk(v, K) are simplicial complexes (flag complexes in fact). It remains to see that if a collection of vertices in Lk(v, L) spans a simplex σ in Lk(v, K) then σ ⊂ Lk(v, L).

Let {v_{1}, …, v_{m}} be a collection of vertices in Lk(v, L) which spans a simplex σ in Lk(v, K). This simplex determines a cell e_{σ} at the vertex v in K. Since v ∈ L we have e_{σ} ∩ L ≠ ∅, and since each v_{i} ∈ Lk(v, L) we see that all the edges of e_{σ} at the vertex v are contained in L. The local convexity condition implies that χ_{e_{σ}}^{−1}(L) = ⊔ ⊓_{e_{σ}} and so e_{σ} ⊂ L. Thus σ ⊂ Lk(v, L) as required. □

Finally, we show that a finite branched covering of a PE cubical complex of nonpositive curvature with branching locus a finite graph is a PE cubical complex of nonpositive curvature. This will be important in Section 6 as we will need to know that the universal cover of such a finite branched covering is a CAT(0) metric space.

Lemma 5.5. Let b : K̂ → K be a finite branched covering of a PE cubical complex K of nonpositive curvature, and suppose that the branching locus L ⊂ K is a finite graph. Then K̂ is also a nonpositively curved PE cubical complex.

Proof. We know from Lemma 5.3 that K̂ is a finite PE cubical complex. It remains to verify that the links are all flag complexes.

Let v ∈ K̂ be a vertex. First we show that Lk(v, K̂) is a simplicial complex. If not, then there exist simplices σ, τ ⊂ Lk(v, K̂) such that σ ≠ τ yet the boundary of σ equals the boundary of τ. The map b induces a map of links b_{∗} : Lk(v, K̂) → Lk(b(v), K). Since Lk(b(v), K) is a simplicial complex, the collection of vertices of σ is mapped injectively by b_{∗} to a collection of veritces in Lk(b(v), K). Furthermore, we have b_{∗}(σ) = b_{∗}(τ) in Lk(b(v), K).

Now each vertex w ∈ σ determines an edge e = e_{w} at v in K̂. If b(e) ⊄ L then the induced map of links b_{∗} : Lk(e, K̂) → Lk(b(e), K) is an isomorphism. Since Lk(e, K̂) may be identified with Lk(w, Lk(v, K̂)) this implies that the map of links b_{∗} : Lk(v, K̂) → Lk(b(v), K) is locally injective at w ∈ Lk(v, K̂). But this contradicts the fact that σ ≠ τ yet b_{∗}(σ) = b_{∗}(τ).

Thus, the vertex set of σ maps injectively to the vertex set of b_{∗}(σ) ⊂ Lk(b(v), K), and each vertex of b_{∗}(σ) is contained in Lk(b(v), L). By Lemma 5.4 Lk(b(v), L) is a full subcomplex of Lk(b(v), K) and so b_{∗}(σ) ⊂ Lk(b(v), L).

So b_{∗} maps both σ and τ injectively to the simplex b_{∗}(σ) ⊂ Lk(b(v), L). But L is a finite graph, and so b_{∗}(σ) is either empty or a single point. Thus σ = τ and this contradiction shows that Lk(v, K̂) is a simplicial complex.

Finally, we show that Lk(v, K̂) is a flag complex. Let {v_{1}, …, v_{n}} be a collection of pairwise adjacent vertices in Lk(v, K̂). Since Lk(b(v), K) is a simplicial complex, this collection maps bijectively to the collection {b_{∗}(v_{1}), …, b_{∗}(v_{n})} of pairwise adjacent vertices in Lk(b(v), K). Now, since Lk(b(v), K) is a flag complex, the {b_{∗}(v_{1}), …, b_{∗}(v_{n})} spans an (n − 1)-simplex. This determines an n-cell based at b(v) in K. This “lifts” to a cell of K̂ which is based at v. Thus the original collection spans an (n − 1)-simplex in Lk(v, K̂) as required. □

It appears that more subtle conditions are necessary in order to preserve nonpositive curvature when the branching locus contains cells of dimension 2 or higher. We are currently investigating this question.

§6. The Example

In this Section we construct a PE cubical complex Y and a continuous map f : Y → S^{1} satisfying the conditions in the following theorem.

Theorem 6.1. There exists a piecewise euclidean cubical complex Y and a continuous map f : Y → S^{1} with the following properties.

(1) The image of ϕ = f_{∗} : π_{1}(Y) → π_{1}(S^{1}) is of finite index in π_{1}(S^{1}).

(2) The universal cover X = Ỹ is a CAT(0) metric space.

(3) X has no embedded flat planes.

(4) The map f lifts to a ϕ-equivariant Morse function f̃: X → ℝ whose ascending and descending links are all homeomorphic to S^{2}.

First we show that Theorem 1.1 follows from Theorem 6.1 above.

Proof of Theorem 1.1. Part (1) implies that there is a short exact sequence of groups

1 → H → G → ℤ → 1

where G = π_{1}(Y), and ℤ = ϕ(G) is a finite index subgroup of π_{1}(S^{1}).

Now G acts freely, properly, cocompactly by isometries on the CAT(0) metric space X by property (2). Property (3) and Theorem 2.2 now imply that X is a δ-hyperbolic metric space, and we conclude from Theorem 2.1 that G is a torsion free hyperbolic group.

Theorem 4.7 and part (4) above imply that H is finitely presented but not of type FP_{3}. Finally, if H were hyperbolic, Theorem 2.3 would imply that its Rips complex is a finite

K(H, 1). In particular, H would be of type FP_{3} (in fact H would be of finite type), which contradicts the conclusion above. □

Proof of Theorem 6.1. The construction of the PE cubical complex Y and continuous map f : Y → S^{1} proceeds as follows.

We begin with a PE cubical complex Θ^{3} and a continuous map d : Θ^{3} → S^{1} such that the image of d_{∗} : π_{1}(Θ^{3}) → π_{1}(S^{1}) is a finite index subgroup of π_{1}(S^{1}), and such that d lifts to a Morse function on the universal cover whose ascending and descending links are all isomorphic to the 2-sphere with simplicial structure given as the join of three 0-spheres (i.e. a regular octahedron). The problem with Θ^{3} is that π_{1}(Θ^{3}) will contain many ℤ ⊕ ℤ subgroups, and so is not hyperbolic.

Next we take a branched covering Y of Θ^{3} so that the universal cover of Y is a δhyperbolic metric space. The map f : Y → S^{1} is just defined to be the composition of this branched covering map and the map d : Θ^{3} → S^{1}. If one is careful in the choice of branched covering, the map f will lift to the universal covers to give a Morse function whose ascending and descending links are all homeomorphic to S^{2} (though not necessarily isomorphic to an octahedron).

Here are the details of these steps.

Step I: The complex Θ^{3} and the map d : Θ^{3} → S^{1}

We denote by Θ the graph consisting of two vertices and four edges, obtained from four copies of the unit interval by identifying all the 0-endpoints and identifying all the 1-endpoints. Define a continuous map

c : Θ → S^{1}

so that its restriction to each of two of the edges of Θ is given by x ↦ [x], and its restriction to each of the other two edges is given by x ↦ [1 − x]. Here [x] denotes the equivalence class of x in ℝ/ℤ. Note that c maps both vertices of Θ to the basepoint [0] ∈ S^{1}. We now take the map c on each factor to obtain a continuous map map c^{3} : Θ^{3} → T^{3}.

The linear map l : ℝ^{3} → ℝ : (x, y, z) ↦ x + y + z induces a continuous map of quotient spaces T^{3} → S^{1} which we also denote by l. Composing l with c^{3} above gives our map

d = c^{3} ∘ l : Θ^{3} → S^{1}

Note that Θ^{3} consists of sixty-four 3-cubes, all sharing the same eight vertices. The cubes are glued together along faces, edges, and vertices, and so Θ^{3} has a natural PE cubical complex strucutre. The link of each vertex is an all-right piecewise spherical complex, which is isomorphic to the 3-fold join of a set of four points. Thus, all the links are flag complexes, and so Θ^{3} is a PE cubical complex of nonpositive curvature.

The map c lifts to a map of universal covers c̃ : Θ̃ → ℝ which is easily seen to be a ϕ-equivariant Morse function whose ascending and descending links are all isomorphic to S^{0}. Thus d = c^{3} ∘ l lifts to a map

d̃= l ∘ c̃^{3} : Θ̃^{3} → ℝ^{3} → ℝ

Now the restriction of d̃ to a 3-cube of Θ̃^{3} is just given by the restriction of the linear map l : ℝ^{3} → ℝ to a 3-cube, possibly postcomposed with an integer translation ℝ → ℝ.

Thus we see that d̃ is a ϕ-equivariant Morse function on Θ̃^{3}. Its ascending and descending links are all isomorphic to the 2-sphere with simplicial structure given as the 3-fold join of S^{0}.

Now Θ̃^{3} is a CAT(0) metric space in the induced path metric, and so Theorem 4.7 implies that Ker(d_{∗} : π_{1}(Θ^{3}) → ℤ) is finitely presented but not of type FP_{3}.

However, π_{1}(Θ^{3}) contains ℤ^{3} subgroups and so is not a hyperbolic group. We now modify this example to obtain a cubical complex Y whose fundamental group is hyperbolic.

Step II: The branched covering Y → Θ^{3}

The idea here is to take a suitable branched covering Y → Θ^{3} so that π_{1}(Y) is a hyperbolic group. However, one must be careful, because branched coverings will affect the local link structure, which in turn may affect the Morse theory argument and the finiteness properties of the kernel subgroup. For instance, after branched coverings the ascending links may no longer be 1-connected. This new link structure would give rise to kernels which might not be finitely presented.

We will take a branched covering of Θ^{3} with the follong branching locus L.

L = (0 × 1 × Θ) ∪ (Θ × 0 × 1) ∪ (1 × Θ × 0) ⊂ Θ^{3}

Properties (1) and (2) of Theorem 6.1 hold for all finite branched covers of Θ^{3} over L. After verifying this, we shall construct an explicit finite branched cover of Θ^{3}, and then show that properties (3) and (4) hold for this covering.

Proof of Theorem 6.1 part (1). In this example, the components of the branching locus L are copies of the graph Θ and so contain loops which map onto the circle S^{1} with degree two. A parallel copy ({1/3} × 1 × Θ) of the component (0 × 1 × Θ) of L also contains loops which map by degree two onto S^{1}. This parallel copy is contained in Θ^{3} \ L and its preimage in the branched covering b : Y → Θ^{3} contains a loop which maps onto S^{1} with nontrivial finite degree. This degree gives an upper bound for the index of ϕ(π_{1}(Y)) in π_{1}(S^{1}). □

Proof of Theorem 6.1 part (2). The complex Y is obtained as a branched covering of the nonpositively curved PE cubical complex Θ^{3}. The branching locus L satisfies the conditions in Lemma 5.3 and Lemma 5.5. Thus Y is a PE cubical complex of nonpositive curvature, and so its universal cover X will be a CAT(0) metric space. □

Now we choose an explicit finite branched covering

b : Y → Θ^{3}

keeping properties (3) and (4) in mind. We begin by considering Θ as a labelled, directed graph as indicated in Figure 1. The edges a and b map to the circle by the map x ↦ [x] and they are directed from 0 to 1. The edges c and d map via x ↦ [1 − x], and they are directed from 1 to 0. As above [x] denotes the equivalence class of the real number x in S^{1} = ℝ/ℤ.

Figure 1. The graph Θ

Definition 6.1. Denote by Δ the graph

Θ × {0} ∪ {1} × Θ ⊂ Θ^{2} \ {(0, 1)}

Note that Δ is a graph with three vertices and eight edges. The edges inherit their orientation from Θ. We choose the following labelling (see Figure 2)

x_{0} denotes x × {0} for x ∈ {a, b, c, d}

and

x_{1} denotes {1} × x for x ∈ {a, b, c, d}

Figure 2. The graph Δ

Remark. The fundamental group π_{1}(Δ, (1, 0)) is free of rank six. We can take as basis the loops:

\overline{a_{0}}b_{0} , \overline{a_{0}}\overline{c_{0}} , \overline{a_{0}}\overline{d_{0}} , a_{1}\overline{b_{1}} , a_{1}c_{1} , a_{1}d_{1}

Lemma 6.2. The inclusion Δ ⊂ Θ^{2} \ {(0, 1)} is a homotopy equivalence.

Proof. We exhibit a strong deformation retraction of Θ^{2} \ {(0, 1)} onto Δ.

Note that Θ^{2} can be obtained from four 2-tori as follows. Each torus is subdivided into four squares by two parallel meridians and two parallel longitudes. Two tori are glued together along a pair of parallel meridians or a pair of parallel longitudes. The glueing pattern can be read from Θ^{2} once we view each Θ as a union of two circles identified along a 0-sphere.

Each torus may be described as the quotient of a square in the plane, bounded by the lines y = ±1 and x = ±1, by identifying points which differ by either a horizontal or a vertical translation through two units. Suppose that the origin maps to the vertex (0, 1) ∈ Θ^{2}. Then radial retraction (along geodesics) from each such square minus the origin to its boundary induces the usual strong deformation retraction from a punctured torus to a figure eight graph, and these paste together to give the deformation retraction Θ^{2} \ {(0, 1)} → Δ. □

Recall that the branching locus is

L = (0 × 1 × Θ) ∪ (Θ × 0 × 1) ∪ (1 × Θ × 0)

Definition 6.3. Define three projections

pr_{i} : Θ^{3} \ L → Θ^{2} \ {(0, 1)} (i = 1, 2, 3)

by

pr_{1} : (x, y, z) ↦ (y, z)
pr_{2} : (x, y, z) ↦ (z, x)
pr_{3} : (x, y, z) ↦ (x, y)

Remark. Note that the maps pr_{i} are all continuous onto. Note also that we have well defined homomorphisms

pr_{i∗} : π_{1}(Θ^{3} \ L , (0, 0, 0)) → π_{1}(Θ^{2} \ {(0, 1)} , (0, 0))

We recall that a finite covering may be prescribed by a representation of the fundamental group of the base into a finite symmetric group. We shall choose a representaion of π_{1}(Δ) into the symmetric group S_{5}, and use the homotopy equivalence of Δ with Θ^{2} \ {(1, 0)} given in Lemma 6.2 together with the maps pr_{i} : Θ^{3} \ L → Θ^{2} \ {(1, 0)} to describe a representation of π_{1}(Θ^{3} \ L, (0, 0, 0)) as follows.

Begin by defining a representation

ρ : π_{1}(Δ, (1, 0)) → S_{5}

which maps the free basis as follows

\overline{a_{0}}b_{o} ↦ α , \overline{a_{0}}\overline{c_{0}} ↦ α^{2} , \overline{a_{0}}\overline{d_{0}} ↦ α^{3}
a_{1}\overline{b_{1}} ↦ β , a_{1}c_{1} ↦ β^{2} , a_{1}d_{1} ↦ β^{3}

where α = (2354) ∈ S_{5} and β = (12345) ∈ S_{5}.

The permutations α and β are chosen to satisfy the relation

αβα^{−1} = β^{2} .

Therefore α^{i}β^{j}α^{−i} = β^{2^{i}j}, and so the following commutation relations hold

[α^{i}, β^{j}] = β^{(2^{i}−1)j} for integers i,j.

Note that for i ∈ {1, 2, 3} and j ∈ {1, 2, 3, 4} the commutator [α^{i}, β^{j}] is a 5-cycle (in fact a nontrivial power of β). The corresponding 5-fold covering has the following important property.

Property 1. There are \binom{4}{2} choices of a pair of edges between (0, 0) and (1, 0) in Δ. Each such pair determines a loop (or its inverse) based at (1, 0) and hence an element of π_{1}(Δ, (1, 0)). Likewise, there are six elements of π_{1}(Δ, (1, 0)) determined by pairs of edges between (1, 0) and (1, 1). The first six elements are simple products of the first three basis elements of π_{1}(Δ, (1, 0)), and so are mapped to α^{i} for i ∈ {1, 2, 3} by ρ. The second six are products of the last three basis elements and map to β^{j} for j ∈ {1, 2, 3, 4} under ρ.

One can form 36 commutators by pairing an element from the first six above with an element from the second six. Each such commutator maps to a nontrivial power of β, and so the preimage of the corresponding loop in \overline{Δ} consists of a single component which maps by degree 5 onto the commutator loop. This property will be crucial in establishing parts (3) and (4) of Theorem 6.1.

For each i ∈ {1, 2, 3} we define the homomorphism ρ_{i} to be the composite

π_{1}(Θ^{3} \ L, (0, 0, 0)) →^{pr_{i}} π_{1}(Θ^{2} \ {(0, 1)} , (0, 0)) →^{≅} π_{1}(Θ^{2} \ {(0, 1)} , (1, 0)) →^{ρ} S_{5}

The middle arrow is just a change of basepoint isomorphism. The arrow labelled ρ involves a slight abuse of notation. It is actually the map ρ precomposed with the inverse of the inclusion induced isomorphism π_{1}(Δ, (1, 0)) → π_{1}(Θ^{2} \ {(0, 1)}, (1, 0)).

These homomorphisms ρ_{i} define three different actions of π_{1}(Θ^{3} \ L , (0, 0, 0)) on the set S = {1, 2, 3, 4, 5} and can be combined to define an action on the set S × S × S. We use this homomorphism

τ = (ρ_{1}, ρ_{2}, ρ_{3}) : π_{1}(Θ^{3} \ L , (0, 0, 0)) → S_{125}

to define a 125-fold cover of Θ^{3} \ L. We lift the local metric from Θ^{3} \ L and complete to obtain the branched covering

b : Y → Θ^{3}

Before proving parts (3) and (4) of Theorem 6.1 we give a description of the links of vertices in Y. A vertex v ∈ Y is said to be of type I if b(v) ∈ {(0, 0, 0), (1, 1, 1)} and is said to be of type II otherwise.

Lemma 6.4. Let b : Y → Θ^{3} be the branched covering defined above, and let v ∈ Y be a vertex. Let F be a 0-complex consisting of just four vertices.

(1) If v is of type I, then Lk(v, Y) is isomorphic to the 3-fold join F ∗ F ∗ F.

(2) If v is of type II, then Lk(v, Y) is isomorphic to the join Γ ∗ F, where Γ → F ∗ F is a finite (5-fold) covering in which each of the thirty six loops of the form S^{0} ∗ S^{0} ⊂ F ∗ F has connected preimage in Γ.

Proof. The vertices (0, 0, 0) and (1, 1, 1) do not lie on the branching locus L and so neighborhoods of lifts of these vertices in Y will be isomorphic to the neighborhoods of (0, 0, 0) or (1, 1, 1) in Θ^{3}. Thus, if v ∈ Y is a vertex of type I, then Lk(v, Y) is isomorphic (as a simplicial complex) to the 3-fold join F ∗ F ∗ F.

A vertex v ∈ Y of type II is the lift of a vertex in Θ^{3} which lies on a component of the branching locus L. This component has a deleted neighborhood of the form

Θ × (F ∗ F) × (0, ϵ) .

This lifts to a subset of the finite cover \overline{Θ^{3} \ L} → Θ^{3} \ L of the form

\overline{Θ} × Γ × (0, ϵ) .

On forming the completion one adds a copy of the graph \overline{Θ} which contains the vertex v. Thus, the link of v is of the form

Lk(v, Y) ≅ Γ ∗ F

and so we need only check that Γ → F ∗ F has the appropriate covering properties.

To determine the covering Γ → F ∗ F we consider a specific case. Suppose the vertex v maps to the vertex (0, 0, 1) ∈ Θ^{3}. Note that (0, 0, 1) lies on Θ × 0 × 1 ⊂ L. We view the F ∗ F above as the boundary of an ϵ-neighborhood of (0, 0, 1) in the 2-complex 0 × Θ^{2}. As such, it is a subset of Θ^{3} \ L and so determines a subgroup of π_{1}(Θ^{3} \ L, (0, 0, 0)) up to conjugacy. The conjugacy class depends on choice of path from (0, 0, 0) to the graph F ∗F. The images pr_{i}(F ∗ F) are contractible for i = 2, 3, while pr_{1} maps F ∗ F isomorphically onto the boundary of an ϵ-neighborhood of (0, 1) in Θ^{2} (this is a subset of Θ^{2} \ {(0, 1)}).

Thus the representation ρ induces an action of the subgroup of π_{1}(Θ^{3} \ L, (0, 0, 0)) determined by F ∗ F on the set S × S × S which is trivial on the 2nd ans 3rd factors. Moreover each of the 36 loops of the form S^{0} ∗S^{0} ⊂ F ∗F map to the commutators defined in Property 1 above, and so act on the first factor by 5-cycles. Thus the preimage of F ∗ F in the 125-fold cover \overline{Θ^{3} \ L} → Θ^{3} \ L determined by ρ consists of 25 components of the form Γ → F ∗F. Each component is a 5-fold covering in which all 36 loops S^{0} ∗S^{0} ⊂ F ∗F have connected preimages. The other cases of type II vertices are handled similarly. □

Proof of Theorem 6.1 part (3). We argue by contradiction. Suppose that i : 𝔼 → X is an isometric embedding of a flat plane. First, we shall show that i(𝔼) intersects a lift of the branching locus transversely at certain points. Secondly, a local argument implies that the angle sum about such a point of 𝔼 exceeds 2π, thus contradicting the fact that 𝔼 is a flat plane.

We say that a point x ∈ 𝔼 is a transverse intersection point with a lift of L if there exists an ϵ > 0 such that the ϵ-ball about x in 𝔼 intersects the i-preimage of the lift of L only in the point x. We say that the flat i : 𝔼 → X intersects that lift of L transversally. Step 1: (Existence of transverse intersection point). Denote by L̃ the preimage of L in X. Each 3-cell of X intersects L̃ in three disjoint, nonparallel edges as shown in Figure 3.

Now i(𝔼) is an isometric embedding, and so is not contained in the 1-skeleton of X. Hence, there exists a 3-cell e ⊂ X such that i(𝔼) ∩ e is 2-dimensional. That is, the set χ_{e}^{−1}(i(𝔼) ∩ e) is a nontrivial polygon of intersection of a 2-plane with a standard 3-cube in ℝ^{3}. Here χ_{e} is an admissible characteristic map for e.

Figure 3. Intersection pattern of L̃ on a 3-cube of X

There are two cases to consider. Firstly, i(𝔼)∩(L̃∩e) ≠ ∅. In this case, the configuration in Figure 3 shows us that χ_{e}^{−1}(i(𝔼) must intersect at least one edge of χ_{e}^{−1}(L̃) in a single point p. Moreover, this edge is not contained in the plane of ℝ^{3} defined by χ_{e}^{−1}(𝔼). We claim that x = i^{−1}(χ_{e}(p)) ∈ 𝔼 is a transverse intersection point.

If p is an interior point of the edge of χ_{e}^{−1}(L̃) then it clearly determines a transverse intersection point x ∈ 𝔼. One just works with an ϵ-ball about x, where ϵ is the minimum distance from p to the endpoints of the edge. If p is an endpoint of the edge, then any ϵ < 1 works in the definition of transverse intersection point. We see this as follows.

If x is not a transverse intersection point, then i(𝔼) must intersect the graph L̃ in points arbitrarily close to the vertex χ_{e}(p). Now i : 𝔼 → X is an isometric embedding, and so if i(𝔼) contains two points on an edge of X it must contain the whole edge. By our choices, i(𝔼) does not contain the edge of L̃ ∩ e at χ_{e}(p), and so must contain one of the other three edges of L̃ at the vertex χ_{e}(p). This edge lies in another 3-cell e′ ⊂ X which shares a 2-dimensional face with e. But the union of these two cells develops into ℝ^{3}, and i(𝔼) ∩ (e ∪ e′) develops into a subset which contains the union of a polygon in the first cube and an edge of the second. Since the edge does not lie in the plane of ℝ^{3} defined by the polygon, we contradict the fact that i is an isometric embedding.

Secondly, i(𝔼)∩(L̃∩e) = ∅. In this case, i(𝔼) must intersect an edge of e which contains a vertex v of type I. Developing i(𝔼) in the neighborhood of v we obtain a 2-dimensional hyperplane in the union of 8 unit cubes about the origin in ℝ^{3} as shown in Figure 4. We choose the admissible characteristic maps so that v maps to the origin and the collection of 8 cubes at v develop out to the unit cubes based at (0, 0, 0), whose union is the cube Q of sidelength 2 with vertex set {−1, 1}^{3} ⊂ ℝ^{3}. Up to composition with an isometry of ℝ^{3}, the intersection pattern with L̃ will develop out as shown in Figure 4. (Note that we have just developed a subset of the neighborhood of v.)

It is easy to see from Euclidean geometry that a plane which intersects the union of the coordinate axes within Q must also transversally intersect one of the 6 lines z = 0, y = ±1; y = 0, x = ±1; x = 0, z = ±1 within Q. Now we argue as in the previous case to see that such intersection points determine transverse intersection points x ∈ 𝔼.

Figure 4. Intersection pattern of L̃ on an ℝ^{3} neighborhood of a type I vertex

Step 2: (Local angle computation at transverse intersection point). Suppose that x ∈ 𝔼 is a transverse intersection point. Then the i-preimage of a 3-cell of X which contains i(x) is a convex polygon based at x in 𝔼 (possibly the degenerate polygon {x}). These polygons are isometric to polygons obtained as the intersections of planes with 3-cubes. We wish to estimate a lower bound for the total angle sum about x. It is hard to determine the angle at x in each polygon, but it is much easier if we group polygons together. Here are the details.

Suppose i(x) = p is an interior point of an edge l of L̃. Then each pair of adjacent polygons at x ∈ 𝔼 maps isometrically to the intersection of i(𝔼) with two 3-cells of X which share a common 2-face. This 2-face contains the edge l. Viewing this locally at p we have a plane intersecting a half space in ℝ^{3}. Thus, these two polygons contribute exactly π to the angle sum at x. Now we shall show that there are at least 6 polygons about x in 𝔼. this gives an angle sum of at least 3π at x, contradicting the fact that 𝔼 is a flat plane.

One may define the link of p in X as we did for vertices. Note that Lk(p, X) is a piecewise spherical cell complex, made from right angled spherical bigons. We also have a cell structure defined locally at x, and so have a map of links i_{∗} : Lk(x, 𝔼) → Lk(p, X) whose image does not intersect Lk(p, L̃).

Let v be an endpoint of the edge l. Since v ∈ L̃ Lemma 6.4 (2) implies that Lk(v, X) is isomorphic to Γ ∗ F. Define a map

pr : ⋃{ e : e is a 3-cell of X which contains l } → X

locally, so that it projects each such 3-cell e onto the face which contains v and is perpendicular to l. Then we have a well defined map

Lk(p, X) \ Lk(p, L̃) → Γ ⊂ Lk(v, X)

taking a unit vector u based at p to the unit vector pr_{∗}(u)/|pr_{∗}(u)|. Composing with i_{∗} we map Lk(x, 𝔼) onto a closed circut in Γ.

Now Γ → (F ∗F) is a finite cover of the bipartite graph (Γ∗Γ), and so Γ is also bipartite. By Property 1 each of the 36 minimal length closed circuts in (F ∗F) have connected 5-fold cover preimages in Γ. Thus, the minimal length of a closed circut in Γ is at least 6. Hence, there are at least 3 pairs of adjacent polygons at x ∈ 𝔼 as required.

This is the first reason for our careful choice of covers \overline{Δ} → Δ and \overline{Θ^{3} \ L} → Θ^{3} \ L. Each lift to Γ of each of the 36 loops (S^{0} ∗ S^{0}) ⊂ (F ∗ F) needs to be a nontrivial covering.

The case where i(x) = p is a vertex of L̃ is treated similarly. In this case, there are four edges of L̃ containing p. One can still define the projection map pr locally as above (p = v here). The local picture at p is a little more complicated. An angle sum of π will correspond to the intersection of a half plane (satisfying z ≥ 0) through the origin in ℝ^{3} with a collection of four unit cubes based at the origin in the upper half space (z ≥ 0). The point p maps to the origin and two of the edges of L̃ map to the y-axis.

The hyperplane will typically intersect three of the four cubes, and so the preimage in 𝔼 may correspond to a sequence of three polygons about x. However, this still just corresponds to a path of length 2 in the graph Γ ⊂ Lk(p, X), and we still get a local angle sum of at least 3π as above. □

Proof of Theorem 6.1 part (4). The branched covering map b : Y → Θ^{3} restricts to give a local isometry on each 3-cube of Y. Therefore, the composite f = db lifts to give a ϕ-equivariant Morse function

f̃: X → ℝ .

It remains to check that the ascending and descending links of f̃ are all 2-spheres.

The local picture of f̃ at a vertex of type I in X is the same as the local picture of d at a vertex of Θ^{3}, and so the ascending and descending links are all isomorphic to the 3-fold join of 0-spheres.

To understand the local picture of f̃ at a vertex of type II we note that f is defined as the composite of d with the branched covering map b. The link of type II vertices is described in Lemma 6.4 (2). The ascending/descending links of d̃ are octahedra

S^{0} ∗ S^{0} ∗ S^{0} ⊂ F ∗ F ∗ F

These lift to X to become subcomplexes of the links F ∗ Γ. By Lemma 5.8 (2), each of the loops of the form S^{0} ∗ S^{0} ⊂ F ∗ F has connected preimage in Γ. Thus the ascending/descending links of f̃ at type II vertices are 2-spheres of the form S^{0} ∗ S^{1} where the simplicial structure on S^{1} is defined as the boundary of a 20-gon.

Note that if the loops S^{0} ∗ S^{0} ⊂ F ∗ F had disconnected preimages in Γ then the ascending/descending links of f̃ would be the suspension of a disconnected graph, and so would have nontrivial π_{1}. This is the second reason for our careful choice of covers \overline{Δ} → Δ and \overline{Θ^{3} \ L} → Θ^{3} \ L. □

This finishes the proof of Theorem 6.1 □

§7. Questions

There appears to be an interesting interplay between the geometry and topology in this example. On the one hand, one needs a lot of branching to ensure that the universal cover of the cubical complex Y is δ-hyperbolic. On the other hand, branched covering affects the local link structure of these cubical complexes. One may end up with a Morse function whose ascending/descending links are not 0- or 1-connected. In such cases, the kernel H may not be finitely presented. One needs to keep this in mind when considering the following questions.

Question 7.1. Does there exist a hyperbolic group G containing a subgroup H which is finitely presented, of type FP_{n}, but not of type FP_{n+1} for n ≥ 3?

If such groups G exist what is their cohomological dimension? How does the minimal cohomological dimension of such G relate to the integer n?

It is tempting to try and generalize the construction in this paper in order to produce examples of hyperbolic groups G containing finitely presented subgroups of type FP_{3} but not of type FP_{4}. One might consider a map from the direct product of four graphs like Θ to S^{1}, and then take branched coverings over some generically chosen 2-subcomplexes. However, the universal covers of the resulting branched coverings always contain isometrically embeddedd flat planes, and so the groups G are not hyperbolic. M. Bestvina first noticed this phenomenon in Gromov’s example.

The basic problem is that there are always embedded flat planes in the universal cover of a finite branched covering of the standard 4-torus (ℝ^{4}/ℤ^{4}), when the branching locus consists of generically chosen 2-subtori which are parallel to the coordinate 2-subtori. See [Be] for details. We have tried to remedy this situation by choosing more complicated branching loci, so that the universal covers become δ-hyperbolic. However, the local link structure always changes so that some of the ascending/descending links are no longer 1-connected.

Question 7.2. Does there exist a hyperbolic group G containing a finitely presented subgroup H of type FP_{∞} (or of type FP) which is not hyperbolic?

There are many interesting questions concerning the geometry of this example. For example, since the subgroup H is not hyperbolic, it must have an isoperimetric inequality which is at least quadratic. What is the isoperimetric inequality of H?

References

[Be] M. Bestvina, On an example of M. Gromov, Talk given at Max Dehn Seminar, University of Utah, 1993. Version available via anonymous ftp from ftp.math.utah.edu under /u/ma/gersten/MaxDehnSeminar/brady.dvi.

[BB] M. Bestvina and N. Brady, Morse theory and finiteness properties of groups, Preprint, University of Utah, August 1995.

[BM] B. Bowditch and G. Mess, A 4-dimensional Kleinian group, Trans. Amer. Math. Soc. 14 no. 1 (1994), 391–405.

[Br] M. Bridson, On the existence of flat planes in spaces of nonpositive curvature, Proc. Amer. Math. Soc. 123 no. 1 (1995), 223–235.

[Br 1] K. Brown, Cohomology of Groups, GTM 87, Springer-Verlag, New York, 1982. [Br 2] K. Brown, Buildings, Springer-Verlag, New York, 1989.

[Da] M. Davis, Nonpositive curvature and reflection groups, Proceedings of the Eleventh Annual Workshop in Geometric Group Theory, Park City, Utah (Hosted by BYU), 1994.

[Ge] S. Gersten, Subgroups of word hyperbolic groups in dimension 2, to appear in Jour. Lon. Math. Soc.

[GdH] E. Ghys and P. de la Harpe (eds.), Sur les Groupes Hyperboliques d’apres Mikhael Gromov, Prog. Math. vol. 83, Birkhauser, Boston, 1990.

[Gr] M. Gromov, Hyperbolic groups, Essays in Group Theory, M.S.R.I. Publ. 8, Springer, New York 1987 (S. M. Gersten, ed.).

[R] E. Rips, Subgroups of small cancellation groups, Bull. Lon. Math. Soc. 14 no. 1 (1982), 45–47.

[S] H. Short (ed.), Notes on hyperbolic groups, Group Theory from a Geometric Viewpoint (Trieste 1990), World Scientific Publishing, River Edge, NJ, 1991.

Department of Mathematics, Cornell University, Ithaca, NY 14853

   E-mail address: brady@math.cornell.edu