New York Journal of Mathematics

New York J. Math. 11 (2005) 89–95.

Bounded geometry in relatively hyperbolic groups

François Dahmani and AslıYaman

Abstract. If a group is relatively hyperbolic, the parabolic subgroups are virtually nilpotent if and only if there exists a hyperbolic space with bounded geometry on which it acts geometrically finitely.

This provides, via the embedding theorem of M. Bonk and O. Schramm, a very short proof of the finiteness of asymptotic dimension for such groups (which is known to imply Novikov conjectures).

Contents

1. Preliminaries 91 2. Polynomial growth for groups and bounded geometry for horoballs 92 3. Proof of Theorem 0.1 93 References 94

The class of relatively hyperbolic groups is an important class of groups encompassing hyperbolic groups, fundamental groups of geometrically finite orbifolds with pinched negative curvature, groups acting on CAT(0) spaces with isolated flats, and many other examples. It was introduced by M. Gromov in [G1] and developed by B. Bowditch, B. Farb, and other authors (e.g., [Bow, F]). There is now an interesting and rich literature on the subject.

A finitely generated group Γ is hyperbolic relative to a family of finitely generated subgroups G if it acts on a proper complete hyperbolic geodesic space X, preserving a family of disjoint open horoballs {B_{p}, p ∈ P }, finite up to the action of Γ, such that the stabilizers G_{p} of the B_{p}, p ∈ P are exactly the elements of G, and for all p, G_{p} acts cocompactly on the horospheres of B_{p}, and such that the action of Γ on X \ (⋃_{p∈P} B_{p}) is cocompact (see [Bow]).

A space X satisfying the conditions of the definition is referred to as an associated space to Γ. Geometrically, one should think of the complement of the horoballs as the universal cover of the convex core of a geometrically finite hyperbolic manifold

89^{ISSN 1076-9803/05}

Received January 31, 2005.

Mathematics Subject Classification. 20F69.

Key words and phrases. Relatively hyperbolic groups, Margulis Lemma, Bounded geometry, asymptotic dimension.

F. Dahmani acknowledges support of the FIM ETH, Zürich.

A. Yaman acknowledges support of the Institute of Mathematics of the University of Bonn and ETH, Zürich. This work was partially done when she was visiting the ETH.

(or equivalently of the thick part of the manifold for Margulis decomposition), and should think of the horoballs as the covers of the cusps.

In many geometrical examples, the parabolic subgroups of Γ, that is, the elements of the family G, are virtually nilpotent. The main examples are geometrically finite manifolds with pinched negative curvature (one can also mention limit groups [D1], groups with boundary homeomorphic to a Sierpinski curve or a 2-sphere [D2]). If the curvature on the manifold is allowed to collapse to −∞, one can obtain other parabolic subgroups (especially nonamenable ones, see [GP, Prop. 0.3]). The difference between these two cases can be identified.

Let us say that a space X is geometrically bounded if there exists a function f : ℝ_{+} → ℝ_{+} such that, for all R > 0, every ball of radius R can be covered by f(R) balls of radius 1, and every ball of radius 1 can be covered by f(R) balls of radius 1/R. In some sense, the function f measures the volume of balls. Such a function always exists when there is a cocompact action on X.

A difference between a complete simply connected manifold M of pinched negative curvature, and M′ in which the curvature is not bounded below, is that M is geometrically bounded, whereas the volumes of a sequence of balls of same radius in M′ may tend to infinity. This remark generalizes:

Theorem 0.1. Let Γ be a finitely generated group, hyperbolic relative to a family G of finitely generated subgroups. Then, every element of G is virtually nilpotent if and only if there exists a space X associated to Γ that has bounded geometry.

The purpose of this note is to prove this characterization, and explain how, in this case, one can deduce short proofs of significant results. Namely we prove that these groups have finite asymptotic dimension, a property with strong consequences.

The asymptotic dimension of a metric space is a quasiisometric invariant introduced by M. Gromov in [G3]. For an introduction, we refer to [R]. It is denoted asdim(X), for a space X, and is defined as follows: it is an integer, and it is less than n ∈ ℕ if and only if for all d > 0 there exists a covering of X by subsets of uniformly bounded diameter, with d-multiplicity at most n + 1.

Here, the d-multiplicity of a covering is the minimal number m such that each ball of radius d intersects at most m elements of the covering.

For the classical examples, the notion gives what is expected, as for example, for Euclidean and hyperbolic spaces, asdim(ℝ^{n}) = asdim(ℍ^{n}) = n (see [R] or [G3]).

Corollary 0.2. Let Γ a finitely generated group that is hyperbolic relative to a family of virtually nilpotent groups. Then, the asymptotic dimension of Γ is finite.

Shortly after our preprint was first posted on arXiv, D. Osin announced a result in [O] that generalizes ours, for groups hyperbolic relative to a family of groups of finite asymptotic dimension. He uses completely different methods. The interest of our method, we believe, is its simplicity and rapidity, and that we show a general property in the spirit of the Margulis Lemma(s).

Once Theorem 0.1 is established, Corollary 0.2 follows from the embedding theorem of M. Bonk and O. Schramm [BS], stating, in particular, that any geometrically bounded Gromov-hyperbolic geodesic space is quasiisometric to some convex subspace of some hyperbolic space ℍ^{n}. Such a space has asymptotic dimension at most n. Applying this to the space associated to Γ given by Theorem 0.1, we get that Γ acts properly discontinuously by isometries on a space that has finite asymptotic dimension. Therefore, it has finite asymptotic dimension itself.

It is worth noting that G. Yu proved in [Y] the coarse Baum-Connes conjecture for proper metric spaces with finite asymptotic dimension. G. Carlsson and B. Goldfarb [CG] proved the integral Novikov conjecture under such a hypothesis.

In order to prove Theorem 0.1, we make use of a certain space X associated to Γ, that is constructed by B. Bowditch in [Bow] when he proves that certain definitions are equivalent. In this model, we first prove, by a growth argument, that, if the horospheres have polynomial growth for their length metric, then the horoballs are geometrically bounded. Then we prove it for the whole space X, using the cocompactness of the action on the complement.

The converse finds its roots in a claim of M. Gromov [G3, p. 150] titled “Generalised and Weakened Margulis Lemma”. We prove it by giving an upper bound to the growth of the volume of a space X associated to Γ, and deduce that a group acting properly discontinuously on an horosphere must have polynomial growth.

Acknowledgements. We would like to thank to Ilya Kapovich for bringing the question to our attention, and Andrzej Szczepanski for his encouragement. We also thank the referee for his/her remarks.

1. Preliminaries

Let Γ be a finitely generated group. We note by gr the growth function of Γ, i.e., for all R > 0, gr(R) is the cardinality of a ball of radius R. It is a well-known theorem of M. Gromov that a finitely generated group is virtually nilpotent if and only if it has polynomial growth (i.e., gr(R) ≤ CR^{p} for some constants C and p). We formulate this latter condition in a slightly different way in the following lemma:

Lemma 1.1. Given a finitely generated group Γ with a word metric, the following are equivalent: A1) For all ϵ < 1 there is a constant N = N(ϵ) such that all balls B(R) of radius

R can be covered by at most N balls of radius ϵR. A2) The growth of Γ is polynomial.

Before giving the proof let us recall a result that characterises polynomial growth.

Theorem 1.2 (H. Bass [Bas]). A group G has polynomial growth if and only if there exist constants K_{1}, K_{2}, p such that for all R, one has K_{1}R^{p} ≤ gr(R) ≤ K_{2}R^{p}.

Proof of Lemma 1.1. If one assumes A1, then, for ϵ = \frac{1}{2}, there exists N such that for all k and R, one has gr(R) ≤ N × gr(ϵR) ≤ N^{k}gr(ϵ^{k}R). Choose k such that 1 ≤ ϵ^{k}R ≤ 2. One has

(e^{k log(ϵ)})^{\frac{log(N)}{log(ϵ)}} ≤ R^{− \frac{log(N)}{log(ϵ)}} ,

hence N^{k} ≤ R^{− log(N)/ log(ϵ)}. Therefore, gr(R) ≤ gr(2) × R^{− log(N)/ log(ϵ)}, what we wanted.

Conversely if one assumes that A1 is not satisfied, then we claim that there is ϵ ∈ (0, 1), such that for all N > 0, there exists R and a ball B of radius R containing N disjoint ball of radius ϵR/4.

To see this, consider a ball B′ of radius R′ that is a counterexample of A1 (for given ϵ < 1 and N − 1), and let choose R = \frac{R′}{1−ϵ/2}, and B the ball of same center as B′ and radius R. Consider B_{1}, …, B_{N}, N balls of radius ϵR′/2 in B. Assume they are not disjoint. By assumption, the balls of same centers but of radius ϵR′ do not fill B′, we choose x be a point in the complement of this union, in B′. Hence, if two of the balls B_{1} … B_{N} intersect, one can exchange one of them with a ball of same radius centered in x, and this one intersects no other, being at distance at least ϵR′ + 1 from any other center. After at most N − 1 of these moves, one has a family of disjoint balls of radius ϵR′/2 = ϵR × (2 − ϵ)/4 ≥ ϵR/4 in B′, and by the triangle inequality, they all lie in B. This proves the claim.

Assuming that Γ has polynomial growth, it satisfies Bass’ Theorem 1.2. It follows from the claim that gr(R) ≥ (K_{1}R^{p}ϵ^{p}/4^{p}) × (N + 1). On the other hand, gr(R) ≤ K_{2}R^{p}. Therefore, K_{2} ≥ K_{1}(ϵ/4)^{p} × (N + 1). As this is true for all N this yields a contradiction. □

2. Polynomial growth for groups and bounded geometry for horoballs

We recall constructions that can be found in the work of Bowditch [Bow] that associate an hyperbolic horoball to every group G, on which G acts, cocompactly on horospheres. We remark that if G has polynomial growth then this space has bounded geometry.

In the upper half plane model of the hyperbolic plane ℍ^{2} let us note T_{t} = [0, 1] × [t, ∞), for all t ≥ 1.

Let K be a connected graph. Let C(K) be the space K × [1, ∞), with a length metric ρ that induces an isometry between T_{1} and e × [1, ∞), for every edge e. It is shown in [Bow] that C(K) is a proper hyperbolic metric space in which every two rays are asymptotic. Its Gromov boundary consists in a single point a.

For all t ≥ 1, let K_{t} be the horosphere K × {t}, at distance t − 1 from K. Note that K_{t} with its induced metric is isometric to (K, d_{t}) where d_{t} = e^{1−t}d_{1}, and d_{1} is the graph metric of K.

Proposition 2.1. There exists constants A, B > 0 and α, β > 0 depending only on the constant of hyperbolicity of (C(K), ρ) such that for all t ≥ 0 and for all x, y ∈ K_{t}, B exp(βρ(x, y)) ≤ d_{t}(x, y) ≤ A exp(αρ(x, y)).

Proof. The lower bound is classical and true in any horosphere of any hyperbolic space. The upper bound follows from the fact that, if [x, y] is a geodesic segment of K_{t}, then its convex hull in (C(K), ρ) is, by construction, isometric to the region of the upper half plane [0, d_{t}(x, y)] × [t, +∞), where the result is classical. □

We denote by π_{t} the orthogonal projection map on K_{t}. For all t ≥ t′ ≥ 0 and for all x, y ∈ K_{t′} we have d_{t′}(x, y) = exp(t − t′)d_{t}(π_{t}(x), π_{t}(y)). Note that the projection map π_{t} sends the balls of K_{t′} to the balls of K_{t}.

Let G be a finitely generated group given with a preferred set of generators and let K_{G} be its associated Cayley graph. We consider the space C(K_{G}) = C(G).

Proposition 2.2. If G has polynomial growth then C(G) has bounded geometry.

Proof. We identify G with the set of vertices of a Cayley graph K_{G} = K_{1}. We first note that K_{1} also satisfies the property A1 of Lemma 1.1. In fact it suffices to show that A1 is satisfied for ϵ = 1/2, since by iteration each ball B(R) of radius R in the Cayley graph can be covered by at most N(1/2)^{n} balls of radius (1/2)^{n}R, where n is the first integer with (1/2)^{n} < ϵ. Thus N(ϵ) = N(1/2)^{n} gives the result. Now when R ≥ 1 the statement is justified by Lemma 1.1, since B(R) can be entirely covered by N′ balls centered at vertices of the Cayley graph and of radius R/2. When R < 1, B(R) is covered by gr(1) + 1 balls of radius R/2. Thus by setting N(1/2) = max{N′, gr(1) + 1} we prove the claim. Note also that this property is invariant by homothety, thus for all t, (K_{t}, d_{t}) satisfies A1.

Let R be a number, and let R′ be equal to R or 1. We want to cover any ball of radius R′ in C(G) by a controlled number of balls of radius R′/R.

Consider a ball B of radius R′ in C(G). Let K_{t} be an horosphere intersecting it. Then K_{t} ∩ B has diameter at most 2R′ in C(G) and hence is contained in a ball of K_{t} of radius A exp(2αR′) for the metric d_{t}, which is homothetic (with factor exp(t − 1)) to (K_{1}, d_{1}). By the remark above this intersection can be covered by N(R′, R, A, B, α, β) balls (note that this number does not depend on t) of radius B exp(βR′/2R) of (K_{t}, d_{t}), which are contained in balls of C(G) of same center, and radius at most R′/(2R) for the ambiant metric.

To cover the entire ball B by balls of radius 1, it is enough to perform this on 4R′ regularly spaced horospheres (at distance 1/(2R′) from each other) intersecting B. One gets a number depending only on R, of balls of radius R′/R that cover B. □

3. Proof of Theorem 0.1

We recall now some of the results and constructions given by Bowditch in [Bow]. Given a group Γ hyperbolic relative to the family G and a space X associated to Γ, he shows that there is a family of disjoint Γ-invariant, quasiconvex horoballs H_{p} based at parabolic points p ∈ ∂X with following properties:

⋆ There are only finitely many orbits of horoballs in the family, and the elements

of G are exactly the stabilizers of the different horoballs. ⋆ The quotient of an horosphere based at p (i.e., the frontier in X of an horoball

based at p) by the stabiliser of p in Γ is compact. ⋆ The quotient of X\ ⋃_{p} int(H_{p}) by Γ is compact.

The proof of these statements can be found in [Bow, Chapter 6] under Lemma 6.3 and Proposition 6.13. Moreover he proves that there exists another associated space to Γ where the horoballs can be chosen to be isometric to the space C(G) where G is a maximal parabolic subgroup in G ([Bow] Chapter 3, Lemma 3.7 and Theorem 3.8). We will refer for the rest of the paper to this particular space as Bowditch’s space. In general a space X associated to a relatively hyperbolic group Γ can be different from it.

Proof of Theorem 0.1. We first prove that if the parabolic subgroups are virtually nilpotent then Bowditch’s space is geometrically bounded. Indeed, as Γ acts on X\ ∪_{p} int H_{p} cocompactly X has bounded geometry if and only if each horoball in X has uniformly bounded geometry. On the other hand since there are only finitely many orbits of horoballs it suffices to show that they all have bounded geometry. But in this particular space, each horoball H_{p} is isometric to some C(G) where G is the stabiliser of p in Γ, and therefore, by Proposition 2.2, this follows.

We turn to the converse. Given a relatively hyperbolic group Γ we assume that there exists a space X of bounded geometry associated to Γ. Denote its metric by ρ. We consider a family of disjoint Γ-invariant horoballs H_{p} based at parabolic points with the above properties, and we note Σ_{p} their horospheres. By assumption each horoball H_{p} has bounded geometry.

For all parabolic points p, let G_{p} be the parabolic group associated. It acts cocompactly on the horosphere Σ_{p}. Let us consider O_{p} an orbit of G_{p} in Σ_{p}, with a metric d_{p} induced by a word metric on G_{p}. Then (O_{p}, d_{p}) is quasiisometric to G_{p}, and note that it is s-separated, for a certain constant s that can be chosen to be 2, up to rescaling X once and for all (note that being geometrically bounded is preserved by rescaling the metric). Thus to show that G_{p} has polynomial growth it is sufficient to show that (O_{p}, d_{p}) has polynomial growth. It is classical that the distortion of O_{p} in X is at least exponential, since it lies on a horosphere.

Let x_{0} ∈ O_{p}, and let us consider R > 0 and B_{R}(x_{0}) the ball of radius R of X centered at x_{0}. Let f(R) be the cardinality of O_{p} ∩ B_{R}(x_{0}). By minoration by an exponential of distances on the horosphere, one deduces that O_{p} ∩ B_{R}(x_{0}) contains a ball of (O_{p}, d_{p}) of center x_{0} and radius A exp(αR), where A and α depend only on X and H_{p}. Let us denote gr_{O} the growth function of O_{p}. Thus one has gr_{O}(A exp(αR)) ≤ f(R).

We now remark that the definition of bounded geometry allows one to map a N-regular tree T on an 1-dense image in X by a 2-Lipschitz map π : T → X, where N is the constant required to cover a ball of radius 2 by balls of radius 1. Indeed it suffices to map the neighbors of a vertex v of the tree to the centers of balls of radius 1 covering a ball of radius 2 centered at π(v).

The growth function of a N-regular tree is exponential. Therefore, the number of disjoint balls of radius 1 that belong to a ball of radius R in X is bounded exponentially depending only on R and X, and hence the function f is at most an exponential, since it counts 2-separated elements in the ball B_{R}(x_{0}). Let us say that f(R) ≤ B exp(β(R)). From this and the fact gr_{O}(A exp(αR)) ≤ f(R), one computes that gr_{O}(t) ≤ B(t/A)^{β/α}, which is polynomial. □

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Labo. E. Picard, Univ. P.Sabatier, 118 Route de Narbonne, F-31062 Toulouse, France. dahmani@picard.ups-tlse.fr http://picard.ups-tlse.fr/˜ dahmani/

IHES Le Bois-Marie, 35, route de Chartres F-91440, Bures-sur-Yvette, France yaman@ihes.fr http://www.ihes.fr/˜ yaman/

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