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Local homology properties of boundaries of groups

Mladen Bestvina

August 16, 1995

0 Introduction

In this paper we formalize the concept of the boundary of a group (Definition 1.1). Even though a group might have different boundaries, certain global and local homological invariants of the boundary are determined by the cohomological invariants of the group. In particular, we show (Theorem 2.8) that the boundary of a Poincaré duality group is a homology manifold.

1 Boundaries of Groups

1.1 Z-structures on groups

Recall that a compact metrizable space is a Euclidean retract (or ER) if it can be embedded in some Euclidean space as its retract, or equivalently if it is finite dimensional, contractible, and locally contractible. A closed subset Z of a Euclidean retract X̃ is said to be a Z-set if any of the following equivalent conditions hold:

• There is a deformation h_{t} : X̃ → X̃ with h_{0} = id and h_{t}(X̃) ∩ Z = ∅ for t > 0.

• For every ϵ > 0 there is a map f : X̃ → X̃ which is ϵ-close to the identity and whose image misses Z.

∗I wish to thank the mathematics department at Melbourne University for their hospitality and support while some of the research leading to this manuscript was conducted.

I also thank the NSF for its continuing support.

• For every open set U ⊂ X̃ inclusion U \ Z ↪ U is a homotopy equivalence.

Definition 1.1. Let G be a group. A Z-structure on G is a pair (X̃, Z) of spaces satisfying the following four axioms.

1. X̃ is an ER.

2. Z is a Z-set in X̃.

3. X = X̃ \Z admits a covering space action of G with compact quotient.

4. The collection of translates of a compact set in X forms a null-sequence in X̃, i.e. for every open cover U of X̃ all but finitely many translates are U-small.

A space Z is a boundary of G if there is a Z-structure (X̃, Z) on G.

It is an immediate consequence of the axioms that the orbit of each point in X accumulates on all of Z, and that for every z ∈ Z, every neighborhood Ũ of z in X̃, and every compact set K ⊂ X there is g ∈ G such that g(K) ⊂ Ũ.

Similar axioms have been considered in the past, notably in attempts to establish the Novikov conjecture [FH81], [CP], [FW].

In many examples the G-action on X extends to an action on X̃. It would seem natural to add this requirement to the above definition (as in the articles quoted above), but we haven’t found a use for it in the present work. Examples 1.2. (i) Every torsion-free word hyperbolic group G admits a Zstructure in which X is the Rips complex of G and Z is the Gromov boundary ∂G [Gro87]. The verification of the axioms in this case is in [BM91].

(ii) If G admits a covering space action by isometries with compact quotient on a CAT(0)-space X, then G admits a Z-structure via compactification by equivalence classes of geodesic rays. In particular, fundamental groups of nonpositively curved closed Riemannian manifolds admit a Z-structure.

(iii) In many cases (braid groups, S-arithmetic groups, Baumslag-Solitar groups, …) it is possible to give ad hoc constructions of structures satisfying (1)-(3) (but not always (4)). We are not aware of any group G that admits a finite K(G, 1) and does not have such a weak Z-structure.

(iv) More concretely, the Baumslag-Soltar group

BS(1, 2) =< x, t|t^{−1}xt = x^{2} >

has a Z-structure with Z homeomorphic to the reduced suspension of the Cantor set (=“Cantor-Hawaiian earring”). The universal cover of the presentation 2-complex is contractible and it can be viewed as the union of “sheets” homeomorphic to the plane. Each sheet can be compactified by adding a circle, and so that any two sheets share exactly one point at infinity. The compactification of sheets is not the usual compactification that results from viewing each sheet as a copy of the hyperbolic plane, but can be constructed from it by blowing up the special point on the circle (the parabolic fixed point) to an arc and collapsing the complementary arc to a point. Another way to construct such a compactification is to identify each sheet with the Euclidean plane so that the usual (CAT(0)) compactification satisfies Axiom 4. A variation of this construction is described in Example 3.1.

(v) If G is a torsion-free geometrically finite Kleinian group in SO(n, 1) with no parabolics, then (ℍ^{n} ∪ S^{n−1}, Λ) is a Z-structure on G, where Λ denotes the limit set of G. Torsion-free uniform lattices in Lie groups admit a Z-structure by (ii). Torsion-free nonuniform lattices in rank 1 Lie groups admit Z-structure in which Z is a Sierpinski space. One blows up each parabolic fixed point on the sphere at infinity to the (n−1)-ball of directions leaving the point.

(vi) Benakli [Ben] has constructed groups whose boundaries are the Sierpinski curve and the Menger curve respectively. In [Dra] Dranishnikov constructs groups (finite index subgroups of Coxeter groups) with boundary n-dimensional Menger compactum or alternatively a space whose cohomological dimension depends on the ground field! See also [BM91] and [Bes93].

In practice, the most difficult axioms to verify are (1) and (2). The following fact (Proposition 2.1 from [BM91]) is useful.

Lemma 1.3. Let (X̃, Z) be a pair of finite-dimensional compact metrizable spaces with Z nowhere dense in X̃, and such that X = X̃ \ Z is contractible and locally contractible, and the following condition holds:

(∗) For every z ∈ Z and every neighborhood Ũ of z in X̃ there is a neighborhood Ṽ of z contained in Ũ such that

Ṽ \ Z ↪ Ũ \ Z

is null-homotopic.

Then X̃ is an ER and Z is a Z-set in X̃.

Lemma 1.4. If (X̃, Z) is a Z-structure on G, and if K is a finite K(G, 1), then there is a natural Z-structure (K̃ ∪ Z, Z) on G, where K̃ denotes the universal cover of K.

Proof. The topology on K̃∪Z comes from taking the closure of the diagonally embedded K̃ in (K̃∪∞)×X̃ (the first factor is the one point compactification of K̃, and the map to the second factor is any G-map K̃ → X ↪ X̃). That this is a Z-structure on G follows from Lemma 1.3.

Therefore, we may always assume without loss of generality that X is a cell complex, and that the G-action is cellular.

Similarly, one can show that if two groups G and G′ with finite Eilenberg-Mac Lane spaces are quasi-isometric, and Z is a boundary of G, then Z is also a boundary of G′.

1.2 Global invariants

Fix a PID 𝕃. Throughout the paper, unless otherwise indicated, all homology and cohomology groups are taken with coefficients in 𝕃. Also, Hom, ⊗, etc. are over 𝕃. We use the Steenrod homology theory (for a review see the next subsection), and the cohomology theory is Čech-Alexander-Spanier.

Fix a Z-structure (X̃, Z) on a group G. For the record, we state an immediate consequence of definitions, the long exact sequence of the pair (X̃, Z), and the fact that X̃ is contractible.

Proposition 1.5. H̃^{q}(Z) ≅ H_{c}^{q+1}(X) ≅ H^{q+1}(G; 𝕃G) and H̃_{q}(Z) ≅ H_{q+1}^{lf}(X).

The shape of Z is determined by G.

Proposition 1.6. If Z_{1} and Z_{2} are boundaries of G, then Z_{1} and Z_{2} have the same shape.

Proof. The sequence of neighborhoods of Z_{i} in X̃_{i} is pro-isomorphic to the sequence of complements of lager and larger compact sets in X.

The following theorem computes the dimension of a boundary in terms of the cohomological dimension of the group. It was stated in [BM91] in the case of hyperbolic groups. The proof is valid in the setting of Z-structures. Recall that when covering dimension dim Z of Z is finite, then dim Z = dim_{ℤ}Z.

Theorem 1.7. Let Z be a boundary of G, and let 𝕃 be a PID. Then

dim_{𝕃}Z = cd_{𝕃}G − 1.

In particular, dim Z = cd G − 1.

1.3 Review of the Steenrod homology theory

Let Z be a compact metrizable space. Embed Z as a Z-set in a compact absolute retract X̃ (e.g. if Z is finite dimensional we can embed it in the boundary of a ball; in general we can embed it in a face of the Hilbert cube) and let X := X̃ \ Z. The proper homotopy type of X depends only on Z. Following Steenrod [Ste40] define the Steenrod reduced homology of Z by

H̃_{i}(Z) := H̃_{i+1}^{lf}(X)

where the right-hand side stands for reduced singular homology based on locally finite chains. Steenrod didn’t shift dimensions. For more thorough expositions and generalizations of this concept see Borel-Moore [BM60], Bredon [Bre67], Milnor [Mil], and Ferry [Fer]. When Z is a polyhedron (or, more generally, an ENR) we can take X̃ to be the cone on Z and we easily see that H̃_{i}(Z) agrees with the usual singular homology.

If Z′ is another compact metric space embedded as a Z-set in an AR X̃′, then any map f : Z → Z′ can be extended to a map F : X̃ → X̃′ so that F^{−1}(Z′) = Z. In particular, the restriction X → X′ = X̃′ \ Z′ is proper and induces a homomorphism between homology groups based on locally finite chains. This homomorphism is declared to be f_{∗} : H̃_{i}(Z) → H̃_{i}(Z′). It depends only on f.

If A is a closed subset of Z, define

H_{i}(Z, A) := H̃_{i}(Z ∪ cA).

It is a pleasant exercise to verify that this homology theory satisfies the usual axioms of Eilenberg-Steenrod.

If z ∈ Z, define local homology groups H_{i}(Z, Z \ {z}) := lim_{B} H_{i}(Z, B) as B ranges over compact subsets of Z \ {z}. More generally, if A ⊆ Z is a closed subspace, define H_{i}(Z, Z \ A) := lim_{B} H_{i}(Z, B) as B ranges over compact subsets of Z \ A.

Proposition 1.8. Let Z be a Z-set in an AR X̃, and let A ⊆ Z be a closed subspace. Choose a basis of neighborhoods {Ũ_{i}} of A in X̃ and let U_{i} = X∩Ũ_{i}. Then

H_{k}(Z, Z \ A) = lim_{→} H_{k+1}^{lf}(U_{i}).

Proof. Let V_{i} := Z ∩ Ũ_{i}. The pair (X̃ \ U_{i}, X̃ \ Ũ_{i}) is excision-equivalent to the pair (Z, Z \ V_{i}) (via the pair (\overline{V}_{i}, ∂V_{i})).

The sequence {H̃_{∗}(X̃ \ Ũ_{i})} is pro-trivial (i.e. sufficiently long compositions of bonding homomorphisms are trivial) since A is a Z-set in X̃ and hence for every i there is j > i so that X̃ \ Ũ_{i} is null-homotopic in X̃ \ Ũ_{j}. It follows that direct sequences

H_{∗}(X̃ \ U_{i}, X̃ \ Ũ_{i}) ≅ H_{∗}(Z, Z \ V_{i})

and

H̃_{∗}(X̃ \ U_{i}) ≅ H_{∗+1}^{lf}(U_{i})

have isomorphic limits.

Remark 1.9. Let Y be a locally compact polyhedron. If 𝕃 is a field, then H_{k}^{lf}(Y) ≅ H_{c}^{k}(Y)^{∗} (the right-hand side is the dual of compactly supported cohomology). More generally, if 𝕃 is a PID, then the Universal Coefficient Theorem provides a short exact sequence

0 → Ext(H_{c}^{k+1}(Y); 𝕃) → H_{k}^{lf}(Y) → Hom(H_{c}^{k}(Y); 𝕃) → 0.

1.4 Rigidity of global cycles in boundaries

Th following proposition imposes restrictions on the nature of the local topology of a boundary of a group.

Proposition 1.10. Suppose that 𝕃 is a countable field, and that Z is a boundary of G.

1. If H^{q+1}(G; 𝕃G) is finite-dimensional, then

H^{q+1}(G; 𝕃G)^{∗} = H_{q+1}^{lf}(X) = H̃_{q}(Z) ↪ H_{q}(Z, Z \ {z})

is injective for every z ∈ Z.

2. If there is a point z ∈ Z such that the local homology module H_{q}(Z, Z \ {z}) is countable, then H^{q+1}(G; 𝕃G) is finite-dimensional.

Proof. Let U_{i} be as in the proof of Proposition 1.8 with A = {z}. (1) First, observe that for every i the inclusion induced homomorphism

H_{c}^{q+1}(U_{i}) → H_{c}^{q+1}(X)

is surjective. Indeed, since dim_{𝕃} H_{c}^{q+1}(X) < ∞, there is a compact set K ⊂ X such that each element of a basis of H_{c}^{q+1}(X) has a representative cocycle with support in K. By Axiom (4) of Definition 1.1, there is a group element g ∈ G such that g(K) ⊂ U_{i}. Since g induces an isomorphism of H_{c}^{q+1}(X), it follows that g(K) contains the support of representative cocycles of a (possibly different) basis of H_{c}^{q+1}(X), and the claim is established.

By taking duals, we see that H_{q+1}^{lf}(X) → H_{q+1}^{lf}(U_{i}) is injective for each i. Taking the limit as i → ∞ implies (1).

(2) Consider the inverse sequence {H_{c}^{q+1}(U_{i})}_{i} of vector spaces. Either for every i there is j > i such that Im[H_{c}^{q+1}(U_{j}) → H_{c}^{q+1}(U_{i})] is finitedimensional or the direct limit of the sequence

{Hom(H_{c}^{q+1}(U_{i}), 𝕃)}_{i}

is uncountable. This direct limit can be identified with H_{q}(Z, Z \ {z}) by Proposition 1.8 and Remark 1.9 and thus the former statement must hold. In particular, we can find i such that s := dim_{𝕃} Im[H_{c}^{q+1}(U_{i}) → H_{c}^{q+1}(X) = H^{q+1}(G; 𝕃G)] < ∞. If dim_{𝕃} H_{c}^{q+1}(X) > s, then choose linearly independent classes x_{1}, x_{2}, ⋯, x_{s+1} ∈ H_{c}^{q+1}(X). There is a compact set K ⊂ X such that each x_{k} has a representative cocycle with support in K. Let g ∈ G be a group element such that g(K) ⊂ U_{i}. By the choice of s, the g-translates of the x_{k}’s are linearly dependent in H_{c}^{q+1}(X), contradicting the fact that g induces an isomorphism of H_{c}^{q+1}(X).

Thus, in the presence of finite dimensionality of global homology of Z, all global cycles are “hung up” on each point of Z, i.e. the support of a nontrivial global class cannot be a proper subset of Z.

The following corollary was proved in [BM91] for Gromov boundaries of hyperbolic groups. The proof there uses the dynamics of the action of the group on the boundary and does not apply in the nonhyperbolic setting.

Corollary 1.11. Let 𝕃 be a field. Suppose that some H^{q+1}(G; 𝕃G) is nontrivial and finite-dimensional over 𝕃. Then Z does not contain a cut point, that is, Z cannot be written nontrivially as the wedge of two compact spaces.

Proof. Suppose that Z = A ∪ B and A ∩ B = {z}, with A and B compact and not singletons. Then H̃_{q}(Z) ≅ H̃_{q}(A) ⊕ H̃_{q}(B) and thus one of H̃_{q}(A) or H̃_{q}(B), say H̃_{q}(A), is nontrivial. But then cycles in H̃_{q}(A) are not hung up on points in B \ {z}.

Remark 1.12. In all examples known to the author it is true that if dim_{𝕃} H_{c}^{q+1}(X) < ∞, then H̃_{q}(Z) → H_{q}(Z, Z \ {z}) is an isomorphism for every z ∈ Z. In particular, the local homology sheaf H_{q} of Z is constant. (See [Bre67]; to an open set V ⊂ Z the sheaf H_{q} assigns H_{q}(Z, Z \ V); the stalk over z ∈ Z is H_{q}(Z, Z \ {z})). That this is the case would be a consequence of the regularity of the compactly supported cochain complex of X (see Definition 2.5).

Remark 1.13. There is an analog of Proposition 1.10 where one assumes only that 𝕃 is a countable PID, and that H^{k}(G; 𝕃G) is finitely generated as an 𝕃-module.

(1) goes through with no changes. By construction, H_{c}^{q+1}(U_{i}) → H_{c}^{q+1}(X) is a split surjecton. This implies that

Hom(H_{c}^{q+1}(X), 𝕃) → Hom(H_{c}^{q+1}(U_{i}), 𝕃)

and

Ext(H_{c}^{q+1}(X), 𝕃) → Ext(H_{c}^{q+1}(U_{i}), 𝕃)

are injective, and then Remark 1.9 and a diagram chase show that

H_{q+1}^{lf}(X) → H_{q+1}^{lf}(U_{i})

is also injective.

(2) is more technical. A theorem of Mitchell [Mit79] states that if 𝕃 is a countable PID, and if {P_{i}}_{i} is an inverse sequence of countable 𝕃-modules, then the following two statements are equivalent:

• For every i there is j > i so that Im[P_{j} → P_{i}] is finitely generated.

• lim Hom(P_{i}, 𝕃) and lim Ext(P_{i}, 𝕃) are countable.

Applying this to the sequence {H_{c}^{q+1}(U_{i})}_{i} we see that either for every i there is j > i so that Im[H_{c}^{q+1}(U_{j}) → H_{c}^{q+1}(U_{i})] is finitely generated or one of lim Hom(H_{c}^{q+1}(U_{i}), 𝕃) and lim Ext(H_{c}^{q+1}(U_{i}), 𝕃) is uncountable. In the first case we conclude that every finitely generated submodule of H^{q+1}(G; 𝕃G) embeds into a fixed finitely generated module. For example, when 𝕃 = ℤ, this implies that H^{q+1}(G; ℤG) is the direct sum of a finite abelian group and a subgroup H of a finite-dimensional vector space over the rationals (and there are also restrictions on the kind of subgroups; there must be a finitely generated group of automorphisms of H such that every element of H can be transformed into a fixed lattice by one of the automorphisms in the group; e.g. if rk H = 1, then H ≅ ℤ[r_{1}, ⋯, r_{m}] for some r_{1}, ⋯, r_{m} ∈ ℚ). In the second case we conclude, by Remark 1.9, that either H_{q}(Z, Z \ {z}) or H_{q−1}(Z, Z \ {z}) is uncountable.

The dual statement to Proposition 1.10(1) about cohomology goes through for PID’s with no change.

Proposition 1.14. If H^{q+1}(G; 𝕃G) is finitely generated as an 𝕃-module and q > 0, then each class in H^{q}(Z) is carried by every nonempty open set U, that is,

H^{q}(Z, Z \ U) → H^{q}(Z)

is onto. In particular, if H^{q+1}(G; 𝕃G) and H^{r+1}(G; 𝕃G) are finitely generated, q, r > 0, and u ∈ H^{q}(Z), v ∈ H^{r}(Z), then u ∪ v = 0.

If H^{q+1}(G; 𝕃G) is not finitely generated, then for every nonempty open set U ⊂ Z the image of H^{q}(Z, Z \ U) → H^{q}(Z) contains an isomorphic copy of every finitely generated submodule of H^{q+1}(G; 𝕃G). In particular, if 𝕃 is a field, this image is infinite-dimensional.

Proof. Let h_{t} : X̃ → X̃ be a small deformation of the identity to a map that misses Z. Use a deck transformation g ∈ G to translate cocycles representing a finite subset of H_{c}^{q+1}(X) close to a point in U. The inverse image under h_{t} of the compact supports of these cocycles intersects Z in a subset of U and represents the carrier of the g-translates of the finite collection in H_{c}^{q+1}(X) ≅ H̃^{q}(Z).

Proposition 1.14 immediately yields Corollary 1.11 in the more general situation when 𝕃 is a PID. Example 1.15. The Cantor-Hawaiian earring (=the reduced suspension of the Cantor set) is a boundary of BS(1, 2). It is not locally connected, and has a cut point.

Amalgamated product of two surface groups over ℤ has a natural boundary (coming from the CAT(0) structure) made up of circles. In particular, there are separating arcs. Collapse one to obtain a boundary with a cut point. This construction is not equivariant.

Corollary 1.16. For every nonempty open set U ⊂ Z

dim_{𝕃}(U) = dim_{𝕃}(Z)(= cd_{𝕃}G − 1).

Proof. If n = cd_{𝕃}(G) then H̃^{n−1}(Z) = H^{n}(G; 𝕃G) ≠ 0. Thus every U carries a cohomology class in H^{n−1}(Z) and its dimension (over 𝕃) cannot be smaller than n − 1.

1.5 Axiom H

Consider the following axiom. A sequence U_{1} ⊃ U_{2} ⊃ ⋯ of open sets in X is basic for z ∈ Z if there is a sequence W̃_{1} ⊃ W̃_{2} ⊃ ⋯ of neighborhoods of z ∈ X̃ forming a basis at z so that sequences {W̃_{i} ∩ X} and {U_{i}} are cofinal in each other.

Axiom H. For every z ∈ Z there is a basic sequence {U_{i}} such that for every n > 1 and every compact set K ⊂ X there exists g ∈ G so that

• g(U_{1} ∪ K) ⊂ U_{n} and

• g(U_{n}) ⊃ U_{m} for some m > n.

The letter H stands for “hyperbolic” (see Proposition 1.18). For example, Axiom H fails for the standard boundary of ℤ^{n} for n > 1.

Proposition 1.17. Let Z be a boundary of G and assume Axiom H. If 𝕃 is a countable field, q ≥ 0 and z ∈ Z, then one of the following holds.

1. The natural map H_{q}(Z) → H_{q}(Z, Z \ {z}) is an isomorphism and the two vector spaces are finite-dimensional.

2. H_{q}(Z, Z \ {z}) is uncountable.

Proof. Let {U_{i}} be a basic sequence at z as in Axiom H. Either there is n > 1 so that V = Im[H_{c}^{q+1}(U_{n}) → H_{c}^{q+1}(U_{1})] has dimension s < ∞ or H_{q}(Z, Z \ {z}) = lim H_{c}^{q+1}(U_{i}) is uncountable. We show that (1) is implied by the former possibility.

We may assume that n is chosen so that s is as small as possible. By the argument of Proposition 1.10 we conclude that dim H_{c}^{q+1}(X) ≤ s (for otherwise s + 1 linearly independent classes could be translated into U_{n} and would yield s + 1 linearly independent classes in V) and that H_{c}^{q+1}(U_{1}) → H_{c}^{q+1}(X) maps V onto H_{c}^{q+1}(X). We now claim that V maps isomorphically onto H_{c}^{q+1}(X). Indeed, if not, there is a compact set K ⊂ X such that inclusion induced homomorphism V → H_{c}^{q+1}(U_{1}∪int K) has nontrivial kernel and image of dimension < s. By Axiom H, there is g ∈ G and m > n such that U_{m} ⊂ g(U_{n}) ⊂ g(U_{1} ∪ int K) ⊂ U_{1} and hence H_{c}^{q+1}(U_{m}) → H_{c}^{q+1}(U_{1}) has image of dimension < s contradicting the choice of n.

If i > n is given, by Axiom H there is j > i such that H_{c}^{q+1}(U_{j}) → H_{c}^{q+1}(U_{i}) has image of dimension ≤ s (since we can interpolate inclusion g(U_{n}) ⊂ g(U_{1}) into U_{j} ⊂ U_{i}). But this image maps onto H_{c}^{q+1}(X) as above. Thus the sequence {H_{c}^{q+1}(U_{i})} is pro-isomorphic to H_{c}^{q+1}(X) and (1) follows.

Proposition 1.18. If G is word-hyperbolic and Z the Gromov boundary, then Axiom H holds.

Proof. To explain the idea, we first focus on the case when G is the fundamental group of a closed hyperbolic manifold M and Z is the sphere at infinity of the universal cover M̃. For a given z ∈ Z choose a geodesic ray r in M̃ that converges to z. For each point x on the ray consider the half-space U_{x} determined by the ray [x, z), namely the set of points y in M̃ such that the angle ∠yxz is < π/2. For a sequence of points x_{i} approaching z along r these half-spaces U_{i} := U_{x_{i}} form a basic sequence for z (this wouldn’t be true in Euclidean space). Since the unit tangent bundle T_{1}M of M is compact, we can choose such a sequence of points x_{i} so that the unit direction vectors at x_{i} pointing towards z are close when projected to T_{1}M. In particular, there is a covering transformation g_{i} that carries x_{1} close to x_{i} and the unit direction vector at x_{1} close to the one at x_{i}. If K and U_{n} are given, for all sufficiently large i we will have g_{i}(U_{1} ∪K) ⊂ U_{n} and g_{i}(U_{n}) will contain some U_{m(i)}. Thus we have verified Axiom H in this case.

The general case follows similarly by the “deltafication” of the above argument. Let G be a word-hyperbolic group and Γ its Cayley graph. Recall that by definition of word-hyperbolicity there exists δ such that for every geodesic triangle ΔABC in Γ the side AB is contained in the δ-neighborhood of the union AC ∪BC of the other two sides. In particular, note the following consequence. If Q is a point on a geodesic segment P R and d(P, Q) > 2δ, d(Q, R) > 2δ, then for every S ∈ Γ we have d(Q, S) < max{d(P, S), d(R, S)}. [Proof: Apply the definition to the triangle ΔP RS to conclude that Q is within δ of a point T in P S or RS, say P S. Since d(P, Q) > 2δ it follows by triangle inequality that d(P, T) > δ and hence d(S, Q) ≤ d(Q, T)+d(T, S) ≤ δ + d(T, S) < d(P, T) + d(T, S) = d(P, S)].

Now recall that the Rips complex M̃ is the simplicial complex with vertex set G and a collection of vertices spans a simplex if the pairwise distance between the vertices in the collection is ≤ d, where d is a large integer. The Cayley graph Γ is naturally contained in M̃. A point z in the Gromov boundary is determined by a geodesic ray r which may be taken to be contained in Γ. Instead of unit tangent vectors we will consider pairs of vertices (x, x′) in M̃ at a fixed distance L > 2δ (measured in Γ) from each other. The halfspace U_{(x,x′)} is defined to be the interior of the subcomplex of M̃ spanned by the vertices y in M̃ that are closer to x′ than to x.

Now choose a sequence of vertices x_{1}, x′_{1}, x_{2}, x′_{2}, x_{3}, x′_{3}, ⋯ along r occurring in that order so that d(x_{i}, x′_{i}) = L and d(x′_{i}, x_{i+1}) > 2δ. By passing to a subsequence we may also assume that there are group elements g_{i} with g_{i}(x_{1}) = x_{i} and g_{i}(y_{1}) = y_{i}. Let U_{i} = U_{(x_{i},y_{i})}. By the above fact we have U_{i} ⊃ U_{i+1}.

We now argue that the sequence {U_{i}} is basic for z. This follows from the two claims below.

Claim 1. If y is a vertex that is closer to x_{i} than to x′_{i}, then for all n > i the distance between x_{1} and any geodesic segment [x_{n}, y] does not exceed δ + d(x_{1}, x_{i+1}).

Indeed, let P and Q be points on [x_{n}, y]∪[y, x_{1}] at distance ≤ δ from x_{i+1} and x_{i} respectively. If one of P, Q belongs to [x_{n}, y] we are done, so assume that they both belong to [y, x_{1}]. Since d(x_{i}, x_{i+1}) > 4δ it must be that P is between y and Q and d(P, Q) > 2δ. Now we have d(y, x_{i+1}) ≤ d(y, P) + δ = d(y, Q) − d(P, Q) + δ ≤ d(y, x_{i}) + δ − d(P, Q) + δ < d(y, x_{i}) contradicting the fact established above.

Claim 2. If y is a vertex that is closer to x′_{i} than to x_{i}, then for all n > i the distance between x_{1} and any geodesic segment [x_{n}, y] is at least d(x_{1}, x_{i}) − δ.

Indeed, let P be a vertex of [x_{n}, y] closest to x_{1}. Let R and S be points in [x_{n}, P] ∪ [P, x_{1}] at distance ≤ δ from x_{i} and x_{i+1} respectively. If one of R, S belongs to [P, x_{1}] we are done, so assume that they belong to [x_{n}, P]. Since d(x_{i}, x_{i+1}) > 4δ the triangle inequality shows that R is between P and S and d(P, S) > 2δ. We now have d(y, x_{i}) ≤ δ + d(R, y) = δ + d(S, y) − d(S, R) ≤ δ + d(x_{i+1}, y) + δ − d(S, R) < d(x_{i+1}, y), a contradiction.

Since g_{i}(U_{1}) = U_{i}, Axiom H follows.

Example 1.19. A “dendrite” made up of a null-sequence of 2-spheres with a dense collection of cut points is not an Axiom H boundary, for it has points where local first homology is trivial, and other points (the cut points) where it is nonzero and countable.

2 The Boundary of a Poincaré Duality Group

Definition 2.1. A compact metrizable space Z of finite covering dimension is said to be a homology n-manifold over 𝕃 if H_{∗}(Z, Z\{z}) ≅ H_{∗}(ℝ^{n}, ℝ^{n}\{0}) for all z ∈ Z. Further, Z is a homology n-sphere if in addition H̃_{∗}(Z) ≅ H̃_{∗}(S^{n}).

Definition 2.2. A group G is a Poincaré duality group of dimension n over 𝕃 (or a P D^{n}-group) if it acts freely, properly discontinuously, and cocompactly on a contractible cell-complex Y with H_{c}^{∗}(Y) ≅ H_{c}^{∗}(ℝ^{n}). The induced action of G on H_{c}^{∗}(Y), i.e. the homomorphism ρ from G into the group of units of 𝕃, is the orientation character.

Examples 2.3. Every P D^{n}-group over ℤ is a P D^{n}-group over any field.

There exist groups that are P D^{n} over some fields, but not over others. One way to construct such groups is to start with a closed PL-manifold M^{n−1} that is a homology sphere over some fields but not over others (e.g. a lens space) and then hyperbolize ([Gro87],[DJ91],[CD95]) the suspension of M.

2.1 Regular chain complexes

Fix a group G with a Z-structure (X̃, Z), and let X = X̃ \ Z.

Definition 2.4. Let

\underline{C} = (0 → C_{l} → C_{l−1} → ⋯ → C_{k+1} → C_{k} → 0)

be a chain complex of finitely generated free 𝕃G-modules. The subscript denotes dimension. For every C_{i} choose an 𝕃G-isomorphism ϕ_{i} between C_{i} and the free abelian group on points in finitely many distinct orbits in X (the latter is an 𝕃G-module via the natural action of G). The support of c ∈ C_{i}, denoted Supp(c), is the finite subset of X consisting of those points in the selected orbits that have nonzero coefficient in the expression of ϕ_{i}(c) in the basis.

Note that if Supp′ is the notion of support with respect to different choices of orbits and ϕ_{i}, then there is a constant M > 0 such that for any c any point in Supp(c) is connected to a point in Supp′(c) by a chain of < M fundamental domains, and vice-versa.

Definition 2.5. We say that \underline{C} is regular if for every z ∈ Z and every open set Ũ ⊂ X̃ containing z there exists a smaller open set Ṽ ⊂ X̃ containing z such that for all i, if c ∈ Im(∂ : C_{i+1} → C_{i}) and Supp(c) is contained in Ṽ, then there exists d ∈ C_{i+1} such that c = ∂d and Supp(d) is contained in Ũ.

In light of Axiom (4) of Definition 1.1, the notion of regularity does not depend on the choice of ϕ_{i}’s.

Topological interpretation. Let Y be a CW complex such that G acts on Y freely, cellularly, and cocompactly, and let Y_{0} be an equivariant subcomplex of Y. We can compactify Y as in Lemma 1.4 by Z. To say that the relative cellular chain complex of (Y, Y_{0}) is regular means that if a relative cycle near a boundary point z bounds in all of Y, then it bounds near z. In general, of course, there will be cycles near z that don’t bound at all.

Examples 2.6. (i) Suppose X is a cell complex and G acts on X cellularly. Then the cellular chain complex (C_{i}(X)) is regular. The reason is that c ∈ C_{i} from the definition can be identified with a (reduced) i-cycle near a boundary point z ∈ Z. Since X̃ is locally contractible at z, we can cone off c staying close to z. Further, since Z is a Z-set, we may assume that the cone misses Z, and thus can be interpreted as a chain in C_{i+1} with boundary c.

(ii) If \underline{C} and \underline{D} are homotopy equivalent (and both are free and finitely generated), and one is regular, so is the other.

Proposition 2.7. Let (X̃, Z) be a Z-structure on an orientable Poincaré duality group G of dimension n (over 𝕃). Also assume that X = X̃ \ Z is a cell complex and that G acts cellularly. Then the cellular cochain complex

0 → C_{c}^{0}(X) → C_{c}^{1}(X) → ⋯ → C_{c}^{m}(X) → 0

is regular (where m = dim X).

Proof. First assume that m = n. The two resolutions of the trivial 𝕃Gmodule 𝕃

0 → C_{c}^{0}(X) → C_{c}^{1}(X) → ⋯ → C_{c}^{n}(X) → 𝕃 → 0

and

0 → C_{n}(X) → C_{n−1}(X) → ⋯ → C_{0}(X) → 𝕃 → 0

are homotopy equivalent. Since the cellular chain complex is regular, so is the compactly supported cochain complex.

If m > n, then using the standard cell-trading arguments one trades away cells of dimension > n and constructs a chain complex of finitely generated free 𝕃G-modules

0 → C_{c}^{0}(X) → C_{c}^{1}(X) → ⋯ → C_{c}^{n−2}(X) → Ĉ_{c}^{n−1}(X) → Ĉ_{c}^{n}(X) → 0

homotopy equivalent to the compactly supported cochain complex (shifted m − n places). This complex is a free resolution of 𝕃 and thus is regular since it is homotopy equivalent to the cellular chain complex of X. Thus the compactly supported cochain complex of X is also regular.

2.2 The boundary of a P D^{n}-group

In the proof of Theorem 2.8 we have to assume that the image of the orientation character is finite (i.e. that G is virtually orientable).

Questions: Is there a P D^{n}-group (say over ℚ) such that the image of the orientation character is infinite (or at least has more than two elements)? Is there an orientable ℚ-homology manifold Z and a homeomorphism Z → Z that acts on the fundamental class by doubling it?

Theorem 2.8. Let G be a Poincaré duality group of dimension n over 𝕃 such that the image of its orientation character is finite. If Z is a boundary of G, then Z is a homology n-manifold over 𝕃, and further Z is a homology (n − 1)-sphere over 𝕃.

Proof. Fix a Z-structure (X̃, Z) on G. By passing to a subgroup of finite index, we may assume that G is orientable (i.e. that the orientation character is trivial). Firstly, H̃_{∗}(Z) ≅ H̃_{∗+1}^{lf}(X) (by definition of the Steenrod homology and Axioms (1) and (2) of Definition 1.1) and hence H̃_{∗}(Z) ≅ H̃_{∗}(S^{n−1}) by the Universal Coefficient Theorem.

Let z ∈ Z be an arbitrary point, and choose a nested sequence of basic open sets Ũ_{i} at z in X̃. Let U_{i} = X ∩ Ũ_{i}. We claim that the inverse sequence {H_{c}^{k}(U_{i})}_{i} is pro-trivial when k ≠ n and that it is pro-isomorphic to 𝕃 when k = n. The theorem then follows from Proposition 1.8 and Remark 1.9.

By Axiom (4) of Definition 1.1, for every compactly supported cocycle c in X there are elements g ∈ G that translate c arbitrarily close to z ∈ Z. By assumption, these translated cocycles represent the same class as c. Thus to prove the claim, it suffices to show that whenever x is a compactly supported cocycle in X with support near z that cobounds a compactly supported cochain in X, then it cobounds a compactly supported cochain with support near z. That this is true is the content of Proposition 2.7.

Remark 2.9. Since homology manifolds of dimension ≤ 2 are manifolds [Wil49], Z is homeomorphic to S^{n−1} when n ≤ 3. Thus the above result generalizes a theorem in [BM91]. When n > 3 Z may not be homeomorphic to S^{n−1} (or even locally simply connected). The examples stem from the work of Davis [Dav83], see [DJ91].

3 Questions

3.1 Existence

There doesn’t seem to be a systematic method of constructing boundaries of groups in general, so the following is probably hopeless.

Question: Does every group G with finite K(G, 1) have a Z-structure? Chapman and Siebenmann [CS76] have developed an obstruction theory

for compactifying noncompact I^{∞}-manifolds to compact I^{∞}-manifolds. By a theorem of Edwards, a product of an ER and I^{∞} is homeomorphic to I^{∞}, so the vanishing of Chapman-Siebenmann obstructions for X × I^{∞} is a necessary condition for the existence of a Z-structure. One way to state this condition is to say that X is proper simple homotopy equivalent to the mapping telescope of an inverse sequence of compact polyhedra.

3.2 Uniqueness

In general, a boundary Z is not uniquely determined by G. For example, take the standard Z-structure for ℤ^{n} in which Z = S^{n−1}, and then quotient out a cell-like subset of Z. When n > 3 we can choose the subset so that the decomposition space is not even a manifold. Even more disturbing is the following example. Example 3.1. Let G have a Z-structure (X̃, Z) and fix a proper map h : X → [1, ∞). Compactify X × ℝ by the suspension Z × [−∞, ∞]/ ∼ of Z so that a sequence (x_{i}, t_{i}) converges to [z, τ] if and only if \frac{t_{i}}{h(x_{i})} → τ and (x_{i} → z or τ = ±∞). If h is chosen so that the variation of h over translates of a compact set forms a null-sequence, then this yields a Z-structure on G × ℤ.

Now let f : G → G be an automorphism. Denote by X_{f} the universal cover of the mapping torus of the map X/G → X/G induced by f. There is a natural map X_{f} → ℝ whose point-preimages are copies of X and the covering translation corresponding to f descends to translation by 1. There is a proper homotopy equivalence f̂: X_{f} → X × ℝ which is given by f^{k} on the point-preimage of k. Using f̂ we can pull back the compactification of X × ℝ. To get a Z-structure on the semidirect product G_{f} of G and ℤ via f we need to ensure that orbits of compact sets are null-sequences. This can be done by choosing h to have very small variation near infinity. For example, h ∼ log log(word length) will do.

In particular, if G is a free group of finite rank and f : G → G an automorphism such that G_{f} is word hyperbolic, then the latter group has two very different boundaries, namely the Gromov boundary (which in many examples is the 1-dimensional Menger curve) and the boundary coming from the above construction (homeomorphic to the suspension of the Cantor set).

Questions: If Z_{1} and Z_{2} are boundaries of G, then is there a compactum Z and cell-like maps Z → Z_{1} and Z → Z_{2}? If so, can Z be chosen to be a boundary of G?

Other obstructions to uniqueness come from the work of Bowers-Ruane [BR]. Thus it seems reasonable to restrict our attention to boundaries satisfying Axiom H.

Questions: If Z_{1} and Z_{2} are Axiom H boundaries of G, then is there a compactum Z and cell-like maps Z → Z_{1} and Z → Z_{2}? If so, can Z be chosen to be an Axiom H boundary of G?

Question: If Z is an Axiom H boundary of a word hyperbolic group G, is Z (equivariantly) homeomorphic to the Gromov boundary of G?

3.3 Regularity

In general, regularity fails for arbitrary Z-structures. Let G = BS(1, 2) × ℤ. Then G has boundary equal to the suspension of the Cantor-Hawaiian earring. We may collapse the arc through the cut point, resulting in a cut point. The cut point has uncountable local H_{1}, while other points have trivial local H_{1}. Thus there are compactly supported 2-cocycles supported near the cut point that don’t vanish near the cut point.

But observe that in any Z-structure, any compactly supported 1-cocycle supported near z ∈ Z that cobounds in X also cobounds near z. The same is true for compactly supported n-cocycles provided n = cd_{𝕃}G and dim_{𝕃}H^{n}(G; 𝕃G) < ∞.

Questions: Let G be a hyperbolic group and Z the Gromov boundary. Is every chain complex of finitely generated free 𝕃G-modules regular? Is the compactly supported cochain complex regular?

3.4 Local homology and local connectedness

As was pointed out before, the following two questions have a negative answer in general. The first fails for BS(1, 2) × ℤ, and the second for BS(1, 2). But the author doesn’t know the answer under additional hypotheses, such as

• Z is the Gromov boundary of a hyperbolic group G, or

• Axiom H, or

• G acts on Z as a convergence group.

Question: If H_{c}^{q+1}(X) is finitely generated over 𝕃, is H_{q}(Z) → H_{q}(Z, Z \ {z}) an isomorphism for each z ∈ Z?

The above question is equivalent to the regularity question for the compactly supported cochain complex.

Question: If H^{k}(G; 𝕃G) = 0 for k ≤ m + 1, is Z locally homologically m-connected (i.e. for every z ∈ Z and every neighborhood A of z in Z there is a smaller neighborhood B such that H̃_{i}(B) → H̃_{i}(A) is trivial for i ≤ m)?

4 Dictionary between groups and compacta

There are some striking analogies between group theory and theory of finite dimensional compact metrizable spaces.

Example 4.1. Hopf theorem on number of ends vs. local H_{1}.

There is a theorem of Hopf [Hop43] that states that if G is a finitely generated and infinite group, then H^{1}(G; 𝕃G) is either 0 (G has 1 end), 𝕃 (G has 2 ends), or infinitely generated (G has infinitely many ends).

This can be compared to the elementary fact that if Z is compact and metrizable, and z ∈ Z, then H_{0}(Z, Z \ {z}) is either 𝕃 (if z is an isolated point), infinitely generated (if there is a sequence of distinct components of Z converging to z), or 0 (otherwise).

Example 4.2. Farrell’s theorem on H^{2}(G; ℤG) vs. Whyburn’s theorem about cut points.

Theorem (Whyburn [Why42]). Let Z be a compact metrizable space. Then Z has at most countably many local cut points of order > 2.

Theorem (Farrell [Far74]). Let G be a finitely presented group. Then H^{2}(G; ℤG) is either 0, ℤ, or infinitely generated.

The boundary of a group G with H^{2}(G; ℤG) = ℤ × ℤ (say) would (in the presence of regularity) be a space with all points local cut points of order 3. Example 4.3. Conner-Floyd-Bredon’s characterization of homology manifolds vs. Farrell’s characterization of P D^{n}-groups.

Theorem (Conner-Floyd [CF59], Bredon [Bre67]). If Z is a connected finite-dimensional compact metrizable space, all local homology groups H_{∗}(Z, Z \ {z}; ℤ) are finitely generated, and the local homology sheaves are (locally) constant, then Z is a homology manifold.

Theorem (Farrell [Far75]). If H^{∗}(G; ℤG) is finitely generated as a ℤ-module, then G is a Poincaré duality group over ℤ. Example 4.4. Farrell spectral sequence vs. Bredon spectral sequence.

Assuming that the compactly supported cochain complex of X is regular, the local homology sheaf H_{∗} is constant and equal to H_{∗}(Z) whenever the latter is finite dimensional. The Bredon spectral sequence [Bre67]

E_{2}^{p,q} = H^{p}(Z; H_{−q}) ⇒ H_{−p−q}(Z)

contains the finite dimensional pieces of the limit (except in dimension 0) in the first column. Deleting those, we obtain a spectral sequence very much like the Farrell spectral sequence [Far75]:

E_{2}^{p,q} ≅ H^{p}(G; Hom(H^{q}(G; 𝕃G), 𝕃G))

and converging to 𝕃 in dimension 0 and to 0 in other dimensions.

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Address: Department of Mathematics, University of Utah, Salt Lake City, UT 84112

E-mail: bestvina@math.utah.edu