Growth of relatively hyperbolic groups

We show that a relatively hyperbolic group either is virtually cyclic or has uniform exponential growth.


Introduction
Let G be a finitely generated group, and S a finite generating set. Denote by d S the word metric on G with respect to S and β(G, S, k) the number of elements of G a d S distance at most k from the identity. The exponential growth rate of G with respect to S is ω(G, S) = lim k→∞ (β(G, S, k)) 1/k . Notice that the limit exists due to the submultiplicativity: β(G, S, m + n) ≤ β(G, S, m)β(G, S, n). We say G has exponential growth if ω(G, S) > 1 for some (hence any) finite generating set S; and we say G has uniform exponential growth if ω(G) = inf S ω(G, S) > 1, where S varies over all finite generating sets of G.
J. Wilson constructed groups that have exponential growth but not uniform exponential growth ( [W]). On the other hand, among the following classes of groups exponential growth implies uniform exponential growth: linear groups over fields with zero characteristic ( [EMO]), hyperbolic groups ( [K]), one-relator groups ( [GD]), solvable groups ( [O2]), and geometrically finite groups acting on pinched Hadamard manifolds ( [AN]). In this paper we show that the same is true for relatively hyperbolic groups.
Theorem 1.1. Let G be a relatively hyperbolic group. Then G either has uniform exponential growth or has a finite index cyclic subgroup.
Relatively hyperbolic groups are generalizations of Gromov hyperbolic groups. Typical examples of relatively hyperbolic groups include Gromov hyperbolic groups, fundamental groups of finite volume real hyperbolic manifolds and groups acting properly and cocompactly on spaces with isolated flats ( [HK]). There are 5 different but equivalent definitions for relatively hyperbolic groups: one due to each of Gromov ([?]), B. Farb ([F]) and D. Osin ([O1]), and two due to B. Bowditch ([B]). We shall use B. Bowditch's definition of relatively hyperbolic groups as geometrically finite groups. The main ingredient in our proof is B. Bowditch's theorem on the existence of an invariant collection of disjoint horoballs (see Proposition 2.2).
The usual way for proving uniform exponential growth is as follows: for any finite generating set S of G, find two elements g 1 , g 2 ∈ G with word length bounded independent of S, such that < g 1 , g 2 > is free with basis {g 1 , g 2 }. We shall use the same strategy. Let G be a relatively hyperbolic group and S a finite generating set. For any positive integer n, let S(n) = {g ∈ G : d S (g, id) ≤ n}. Notice that S(n) is also a finite generating set of G, and S(n) ⊂ S(m) if n ≤ m. In particular, S ⊂ S(n) for all n ≥ 1.
We prove uniform exponential growth in two steps: Step 1: There exists some positive integer n 0 with the following property. For any finite generating set S of G, S(n 0 ) contains a hyperbolic element. See Proposition 5.1.
Step 2: There exists some positive integer k 0 with the following property. If a finite generating set S of G contains a hyperbolic element, then there are g 1 , g 2 ∈ S(k 0 ) such that < g 1 , g 2 > is free with basis {g 1 , g 2 }. See Corollary 3.2 and Proposition 4.1.
It should be noted that uniform exponential growth for geometrically finite groups acting on pinched Hadamard manifolds has been established by R. Alperin and G. Noskov ([AN]). Some of our arguments are similar to theirs. Theorem 1.1 was conjectured by C. Drutu in [D].
Notation: For a metric space Y , any subset A ⊂ Y and any r ≥ 0, we denote by N r (A) = {y ∈ Y : d(y, A) ≤ r} the closed r-neighborhood of A. For any p, q ∈ Y , pq denotes a geodesic segment between p and q, although it is not unique in general.

Relatively hyperbolic groups as geometrically finite groups
Here we recall the notion of geometrically finite groups and B. Bowditch's result on the existence of an invariant system of disjoint horoballs (see Proposition 2.2). The reader is referred to [B] for more details.
Suppose that M is a compact metrizable topological space and a group G acts by homeomorphisms on M. We say that G is a convergence group if the induced action on the space of distinct triples is properly discontinuous. Let G be a convergence group on M. A point ξ ∈ M is a conical limit point if there exists a sequence in G, {g n } ∞ n=1 , and two points ζ = η ∈ M, such that g n (ξ) → ζ and g n (ξ ′ ) → η for all ξ ′ = ξ. An element g ∈ G is a hyperbolic element if it has infinite order and fixes exactly two points in M. A subgroup H < G is parabolic if it is infinite, fixes a point ξ ∈ M, and contains no hyperbolic elements. In this case, the fixed point of H is unique and is referred to as a parabolic point. The nontrivial elements in a parabolic subgroup are called parabolic elements. A parabolic point ξ is a bounded parabolic point if its stabilizer Stab(ξ) := {g ∈ G : g(ξ) = ξ} acts properly and cocompactly on M\{ξ}. A convergence group G on M is a geometrically finite group if each point of M is either a conical limit point or a bounded parabolic point.
Definition 2.1. A group G is hyperbolic relative to a family of finitely generated subgroups G, if it acts properly discontinuously by isometries, on a proper geodesic hyperbolic space X, such that the induced action on ∂X is of convergence, geometrically finite, and such that the maximal parabolic subgroups are exactly the elements of G.
By definition a relatively hyperbolic group is a geometrically finite group. A. Yaman ( [Y]) proved that if G is a geometrically finite group on a perfect metrizable compact space M, and the maximal parabolic subgroups are finitely generated, then G is hyperbolic relative to the family of maximal parabolic subgroups. Now let X be a δ-hyperbolic geodesic metric space for some δ ≥ 0. Recall that each geodesic triangle in X is δ-thin, that is, each edge is contained in the δ-neighborhood of the union of the other two edges. Below we shall denote by c = c(δ) a constant that depends only on δ. Let ξ ∈ ∂X. A (not necessarily continuous) function h : X → R is a horofunction about ξ if there are constants c 1 = c 1 (δ), c 2 = c 2 (δ) such that: if x, a ∈ X and d(a, xξ) ≤ c 1 for some geodesic ray xξ from x to ξ, then and h(x) ≤ c for all x ∈ X\B. Note that ξ is uniquely determined by B, and we refer to it as the center of the horoball.
Proposition 2.2. (B. Bowditch, Proposition 6.13 in [B]) Let G be a relatively hyperbolic group, and X a proper hyperbolic geodesic metric space that G acts upon as in Definition 2.1. Let Π be the set of all bounded parabolic points in ∂X. Then Π/G is finite. Moreover, for any r ≥ 0, there is a collection of horoballs B = {B ξ : ξ ∈ Π} indexed by Π with the following properties:

Axes of hyperbolic elements
In this section we study how a hyperbolic element "translates" its "axes".
Let G and X be as in Definition 2.1. That is, X is a proper δ-hyperbolic geodesic space for some δ ≥ 0, G acts properly discontinuously by isometries on X, such that the induced action on ∂X is of convergence and geometrically finite.
Recall that, in a δ-hyperbolic space X, any two complete geodesics that have the same endpoints in ∂X have Hausdorff distance at most 2δ. For a hyperbolic element γ ∈ G, let γ + and γ − be the attracting and repelling fixed points of γ in ∂X respectively. We shall call any complete geodesic with γ + , γ − as endpoints an axis of γ, and denote by A γ the union of all axes of γ. Note that γ may have many different axes and an axis of γ in general is not invariant under γ. However, for any axis c of γ, γ(c) is also an axis of γ and hence the Hausdorff distance between c and γ(c) is at most 2δ.
Proposition 2.2 implies that there is a 200δ-separated G-invariant collection of horoballs B centered at the parabolic points such that Y (B)/G is compact.
Lemma 3.1. There exists a positive integer k 1 with the following property: for any infinite order element γ ∈ G and any Proof. Notice that the action of G on Y (B) is properly discontinuous and cocompact. It follows that there is some integer k 1 ≥ 1 such that for any x ∈ Y (B), the cardinality of {g ∈ G : d(x, g(x)) ≤ 200δ} is less than k 1 . In particular, for any x ∈ Y (B) and any infinite order element γ ∈ G, there is some k, Corollary 3.2. If a finite generating set S of G contains a hyperbolic element, then there is a hyperbolic element γ ∈ S(k 1 ) and some Proof. Let g ∈ S be a hyperbolic element and c an axis of g. It follows from the definition of a horoball that c is not contained in any horoball. Since B is a disjoint collection of horoballs and c is connected, Notice that g k ∈ S(k 1 ) is hyperbolic and c is also an axis of g k .
For a complete geodesic c in X, we define a map P c : X → c as follows: for any Proof. Recall that the Hausdorff distance between c and c ′ is at most 2δ. It follows from triangle inequality that |b , c(a)) ≤ 6.5δ, contradicting the fact that b ≥ a + 8δ.
Lemma 3.4. Let g ∈ G be a hyperbolic element, and c an axis of g. Suppose x ∈ c is a point with d(g(x), x) ≥ 20δ. Then for any integers i < j, the point P c (g j (x)) lies between P c (g i (x)) and g + .
Proof. Denote x i = P c (g i (x)). Note that x i → g + as i → +∞. It suffices to show that for any i, x i lies between x i−1 and x i+1 . Set t = d(g(x), x). Triangle inequality , the other case can be handled similarly. Let y 1 = P c (g(x 1 )). Then y 1 = σ(a) for some a ∈ R. Now Lemma 3.3 applied to σ, g • σ and the projections of g(x), g( Lemma 3.5. Let g ∈ G be a hyperbolic element. Suppose there is a point x ∈ A g with d(x, g(x)) ≥ 200δ. Then |d(y, g(y)) − d(z, g(z))| ≤ 40δ for all y, z ∈ A g .

Free subgroups
The goal in this section is to find two short hyperbolic elements that generate a free group: Proposition 4.1. Let G be a relatively hyperbolic group, and X a proper δ-hyperbolic geodesic metric space that G acts upon as in Definition 2.1. Then there exists a positive integer k 2 with the following property. If S is a finite generating set of G and s ∈ S is a hyperbolic element such that d(s(x), x) ≥ 200δ for some x ∈ A s , then there are g 1 , g 2 ∈ S(k 2 ) such that < g 1 , g 2 > is free with basis {g 1 , g 2 }.

Ping-Pong lemma
The proof of Proposition 4.1 is based on the following Ping-Pong lemma.
Lemma 4.2. Let G be a group acting on a set X, and g 1 , g 2 two elements of G. If X 1 , X 2 are disjoint subsets of X and for all n = 0, i = j, g n i (X j ) ⊂ X i , then the subgroup < g 1 , g 2 > is free with basis {g 1 , g 2 }.
We will apply the Ping-Pong Lemma in the following setting.
Lemma 4.3. Let X be a metric space and g 1 , g 2 isometries of X. Let B i ⊂ A i ⊂ X (i = 1, 2) be subsets of X, and p i : Then the assumptions in the Ping-Pong lemma are satisfied if the following conditions hold: (1) X 1 ∩ X 2 = ∅; (2) g n i (p −1 i (B i )) ⊂ X i for all n = 0, i = 1, 2.
In our case, g i is a hyperbolic element, A i is an axis of g i , B i ⊂ A i is a segment, and p i = P A i .

Two hyperbolic elements
In this section we prove Proposition 4.1. For this, we shall find two short hyperbolic elements and segments in their axes such that the two conditions in Lemma 4.3 are satisfied.
It is easy to see from the δ-thin condition that the Hausdorff distance between yz and y ′ z ′ is at most 4δ. After replacing h by its inverse if necessary, we may assume g and h translate in the "same direction", that is, if both c 1 and c 2 are parameterized from the repelling fixed point toward the attracting fixed point, then b ′ > a ′ (we have b > a by definition).
Since h is a conjugate of g or its inverse, Lemma 3.5 implies that the inequality |d(p, g i (p)) − d(q, h i (q))| ≤ 40δ holds for all i ≥ 1, and all p ∈ A g , q ∈ A h . In particular, |d(y, g i (y)) − d(y ′ , h i (y ′ ))| ≤ 40δ for all i ≥ 1.
For i ≥ 0, let y i = P c 1 (g i (y)) and y ′ i = P c 2 (h i (y ′ )). Lemma 3.4 implies that y i+1 lies between y i and c 1 (+∞). Since d( ) ≤ 2δ + 50δ + 4δ + 2δ = 58δ. Now for any i, 1 ≤ i ≤ k 1 , we have d(y, h −i • g i (y)) = d(h i (y), g i (y)) ≤ d(h i (y), h i (y ′ )) + d(h i (y ′ ), g i (y)) = d(y, y ′ ) + d(h i (y ′ ), g i (y)) ≤ 60δ. Since y ∈ Y (B), it follows from the definition of k 1 that there are i = j, 1 ≤ i, j ≤ k 1 with h −i • g i = h −j • g j . Consequently h j−i = g j−i , contradicting the fact that g and h do not share any fixed points in ∂X. B). There are horoballs B 1 , B 2 ∈ B with y ∈ B 1 and g(y) ∈ B 2 . Note B 1 = B 2 , otherwise g(B 1 ) = B 1 and so g fixes the center of B 1 , contradicting the fact that g is a hyperbolic element. Since B is 200δ-separated and d(y 1 , g(y)) ≤ 2δ, there is some p ∈ c 1 between y and y 1 such that p ∈ Y (B). Now we run the argument from the pervious paragraph using p instead of y.
Proof. We only write down the proof in the case d(A 1 , A 2 ) ≤ 2δ, i = 1 and n > 0, the other cases are similar or simpler.

Existence of hyperbolic elements
In this section we establish the following result, which guarantees the existence of hyperbolic elements with short word length.
Proposition 5.1. Let G be an infinite relatively hyperbolic group and X a δ-hyperbolic geodesic metric space that G acts upon as in Definition 2.1. Then there exists a positive integer n 0 with the following property: for any finite generating set S of G, S(n 0 ) contains a hyperbolic element.
For a finite set of isometries F of a metric space X, and x ∈ X, we let λ(x, F ) = max{d(f (x), x) : f ∈ F }. We use the following result of M. Koubi ([K]).
Proposition 5.2. Let X be a δ-hyperbolic geodesic metric space, and G a group of isometries of X with finite generating set S. If λ(x, S) > 100δ for all x ∈ X, then G contains a hyperbolic element g such that d S (id, g) = 1 or 2.
Proposition 5.1 follows from Proposition 5.2 and the following result.
Lemma 5.3. There exists a positive integer n 0 with the following property. For any finite generating set S of G, the inequality λ(x, S(n 0 )) > 100δ holds for all x ∈ X.