Growth of quotients of groups acting by isometries on Gromov hyperbolic spaces

We show that every non-elementary group $G$ acting properly and cocompactly by isometries on a proper geodesic Gromov hyperbolic space $X$ is growth tight. In other words, the exponential growth rate of $G$ for the geometric (pseudo)-distance induced by $X$ is greater than the exponential growth rate of any of its quotients by an infinite normal subgroup. This result generalizes from a unified framework previous works of Arzhantseva-Lysenok and Sambusetti, and provides an answer to a question of the latter.


Introduction
In this article, we investigate the asymptotic geometry of some discrete groups (G, d) endowed with a left-invariant metric through their exponential growth rate. The exponential growth rate of (G, d), also called entropy or critical exponent, is defined as where B G (R) is the ball of radius R formed of the elements of G at distance at most R from the neutral element e. (Some authors define the exponential growth rate of (G, d) as the exponential of ω(G, d).) The quotient groupḠ = G/N of G by a normal subgroup N inherits the quotient distanced given by the least distance between representatives. The distanced is also leftinvariant. Clearly, we have ω(Ḡ,d) ≤ ω(G, d). The metric group (G, d) is said to be growth tight if ω(Ḡ,d) < ω(G, d) (1.2) for any quotientḠ of G by an infinite normal subgroup N ¡ G. In other words, (G, d) is growth tight if it can be characterized by its exponential growth rate among its quotients by an infinite normal subgroup. Observe that if the normal subgroup N is finite, then the exponential growth rates of G andḠ clearly agree.
The notion of growth tightness was first introduced by R. Grigorchuk and P. de la Harpe [GrH97] for word metrics on finitely generated groups. In this context, A. Sambusetti [Sa02b] showed that every nontrivial free product of groups, different from the infinite dihedral group, and every amalgamated product of residually finite groups over finite subgroups are growth tight with respect to any word metric. In another result, G. Arzhantseva and I. Lysenok [AL02] gave an affirmative answer to the question about growth tightness of word hyperbolic groups posed by R. Grigorchuk and P. de la Harpe [GrH97]. More precisely, they proved that every nonelementary word hyperbolic group is growth tight for any word metric. Recently, W. Yang [Ya] extended this result to non-elementary relatively hyperbolic groups (and more generally to groups with nontrivial Floyd boundary), still for any word metric. On the other hand, it is not difficult to check that the direct product F 1 × F 2 of two free groups of rank at least 2 is not growth tight for the word metric induced by the natural basis of F 1 × F 2 obtained from two free basis of F 1 and F 2 , cf. [GrH97].
Applications of growth tightness to geometric group theory in connection with the Hopf property and the minimal growth of groups can be found in [GrH97,Sa01,AL02,Sa02a,Sa02b,Sa04,CSS04].
Word metrics are not the only natural metrics which arise on groups. For instance, let G be the fundamental group of a closed Riemannian manifold (M, g) with basepoint x 0 . The group G acts properly and cocompactly by isometries on the Riemannian universal cover (M ,g) of (M, g). The distance on G induced by g between two elements α, β ∈ G, denoted by d g (α, β), is defined as the length of the shortest loop based at x 0 representing α −1 β ∈ G = π 1 (M, x 0 ). Every quotient groupḠ = G/N by a normal subgroup N ¡ G is the deck transformation group of the normal coverM =M /N of M . The quotient distanced g onḠ agrees with the distance dḡ onḠ induced by the liftḡ onM of the Riemannian metric g on M (here, we take for a basepoint onM any lift of x 0 ). Furthermore, the exponential growth rate of (Ḡ, dḡ) agrees with the one of the Riemannian cover (M ,ḡ) defined as ω(M ,ḡ) = lim R→+∞ log vol Bḡ(R) R where Bḡ(R) is the ball of radius R inM centered at the basepoint (the limit exists since M is compact). In other words, we have ω(Ḡ, dḡ) = ω(M ,ḡ). Note that the exponential growth rates ofḠ andM do not depend on the choice of the basepoint. By definition, the exponential growth rate of the Riemannian universal cover (M ,g) is the volume entropy of (M, g).
As pointed out in [Sa08], even though the fundamental group of M is a word hyperbolic group in the sense of Gromov, its geometric distance is only quasi-isometric to any word metric. Since the exponential growth rate of the fundamental group of M (hence, apriori, its growth tightness) is not invariant under quasi-isometries, it is not clear how to derive this theorem from its group-theoretical counterpart.
Clearly, this theorem does not extend to nonpositively curved manifolds: the product of a flat torus with a closed hyperbolic surface provides a simple counterexample. In [Sa04], A. Sambusetti asks if growth tightness holds for any Riemannian metric, without curvature assumption, on a closed negatively curved manifold. We affirmatively answer this question in a general way unifying different results on the subject, cf. Theorem 1.3.
The following classical definitions will be needed to state our main result.
Definition 1.2. A metric space X is proper if all its closed balls are compact and geodesic if there is a geodesic segment joining every pair of points of X. Following E. Rips's definition, a geodesic metric space X is δ-hyperbolic if each side of a geodesic triangle of X is contained in the 4δ-neighborhood of the union of the other two sides (we refer to Section 4 for the original definition of δ-hyperbolicity). A geodesic metric space is Gromov hyperbolic if it is δ-hyperbolic for some δ ≥ 0. A group G is elementary if it contains a (finite or infinite) cyclic subgroup of finite index. A group G acts properly on a metric space X if for any compact set K ⊂ X, there are only finitely many α ∈ G such that α(K) intersects K. A group G acts cocompactly on a metric space X if the quotient space X/G is compact.
Let G be a group acting by isometries on a metric space X = (X, | · |) with origin O. The distance on X induces a left-invariant pseudo-metric d on G given by for every α, β ∈ G, where |xy| represents the distance between a pair of elements x, y ∈ X. The notion of exponential growth rate for G = (G, d) extends to the pseudo-distance d and does not depend on the choice of the origin O.
In this article, we consider a non-elementary group G acting properly and cocompactly by isometries on a proper geodesic δ-hyperbolic metric space X. This implies that G is finitely generated. Furthermore, its Cayley graph with respect to any finite generating set is quasi-isometric to X and so is Gromov hyperbolic. Thus, G is a word hyperbolic group in the sense of Gromov. In particular, the exponential growth rate of G is positive for the (pseudo)-metric induced by the distance on X and any word distance since non-elementary hyperbolic groups have non-abelian free subgroups. Note also in this case that the limit-sup (1.1) is a true limit, cf. [Co93].
We can now state our main result. Theorem 1.3. Let G be a non-elementary group acting properly and cocompactly by isometries on a proper geodesic δ-hyperbolic metric space X. Then G is growth tight for the (pseudo)-metric induced by the distance on X.
A quantitative version of this result can be found in the last section. Note that the exponential growth rate of a quotient groupḠ of G can vanish. In this case, the relation (1.2) is clearly satisfied. Theorem 1.3 yields an alternative proof of the growth tightness of nonelementary word hyperbolic groups in the sense of Gromov for word metrics, cf. [AL02]. Indeed, every word hyperbolic group in the sense of Gromov acts properly and cocompactly by isometries on its Cayley graph (with respect to a given finite generating set) which forms a proper geodesic Gromov hyperbolic space.
Since the universal cover of a closed negatively curved n-manifold M is a Gromov hyperbolic space, we recover Theorem 1.1 as well by taking G = π 1 (M ) and X =M . Actually, Theorem 1.3 allows us to extend this result in three directions as it applies to  Obviously, Corollary 1.4 applies when M is diffeomorphic to a closed locally symmetric manifold of negative curvature. One may wonder if the conclusion holds true when M is diffeomorphic to a closed irreductible higher rank locally symmetric manifold of noncompact type. This question finds its answer in Margulis' normal subgroup theorem [Ma91]. Indeed, in the higher rank case, the only normal covers of M are either compact (their exponential growth rate is zero) or are finitely covered by the universal coverM of M (their exponential growth rate agrees with the exponential growth rate ofM ).
In [DPPS11, §5.1], building upon constructions of [DOP00], the authors show that there exists a noncompact complete Riemannian manifold M with pinched negative curvature and finite volume which does not satisfy the conclusion of Corollary 1.4 even forM =M . This shows that we cannot replace the Gromov hyperbolic space X by a relatively hyperbolic metric space in Theorem 1.3.
Finally, we mention that the gap between the exponential growth rates of G andḠ can be arbitrarily small. For free groups, this follows from [Sh99], see also [Ta05,Lemma 3]. Even for word hyperbolic groups in the sense of Gromov, this is also true. Indeed, it has recently been established in [Cou] that the exponential growth rate of the periodic quotient G/G n of a nonelementary torsion-free word hyperbolic group G is arbitrarily close to the exponential growth rate of G, for every odd integer n large enough. Here, G n represents the (normal) subgroup generated by the n-th powers of the elements of G.
The strategy used to prove Theorem 1.3 follows and extends the approach initiated by A. Sambusetti and used in [Sa01,Sa02a,Sa02b,Sa03,Sa04,Sa08,DPPS11]. However, the nature of the problem leads us to adopt a more global point of view, avoiding the use of any hyperbolic trigonometric comparison formula, which cannot be extended in the absence of curvature assumption and without a control of the topology of the spaces under consideration. We present an outline of the proof in the next section.
Acknowledgments. The author is grateful to the referee for his or her detailed report and useful comments, which helped improve the article, as well as for pointing out some pertinent references.

Outline of the proof
In this section, we review the approach initiated by A. Sambusetti and developed in this article.
Let G be a non-elementary group acting properly and cocompactly by isometries on a proper geodesic δ-hyperbolic metric space X. Every quotient groupḠ = G/N by a normal subgroup N ¡G acts properly and cocompactly by isometries on the quotient metric spaceX = X/N . We will assume that ω(Ḡ) is nonzero, otherwise the main result clearly holds. This implies that X is unbounded.
Fix an origin O ∈ X. The left-invariant pseudo-distance d, cf. (1.3), induces a semi-norm || · || G on G given by for every α ∈ G. By definition, a semi-norm on G is a nonnegative function || · || defined on G such that ||α −1 || = ||α|| ||αβ|| ≤ ||α|| ||β|| for every α, β ∈ G. Semi-norms and left-invariant pseudo-distances on a given group are in bijective correspondence. Similarly, we define a seminorm || · ||Ḡ onḠ from the quotient pseudo-distanced. For the sake of simplicity and despite the risk of confusion, we will drop the prefixes pseudoand semi-in the rest of this article and simply write "distance" and "norm".
A direct combinatorial computation [Sa02b, Proposition 2.3] shows that More precisely, the following estimate holds Strictly speaking this estimate has been stated for true distances, but it also applies to pseudo-distances.
If we could construct an injective 1-Lipschitz map we could claim that the R-ball BḠ * Z 2 ,λ (R) ofḠ * Z 2 for d λ injects into an R-ball of G. This would imply that ω(Ḡ * Z 2 , λ) ≤ ω(G). Combined with Proposition 2.1, the main theorem would follow. Here, we do not construct such a nonexpanding embedding, but derive a slightly weaker result which still leads to the desired result.
Fix ρ > 0. LetḠ ρ be a subset ofḠ containing the neutral elementē ∈Ḡ such that the elements ofḠ ρ are at distance greater than ρ from each other and every element ofḠ is at distance at most ρ from an element ofḠ ρ .
Example 2.2. Let G be the free group F 2 = Z * Z endowed with the word metric induced by the canonical generators of the Z factors. For ρ = 5/4, the setḠ ρ is formed of all words of even length.
Thus, the key argument in the proof of the main theorem consists in constructing and deriving the properties of the nonexpanding map Φ, cf. (2.1). This is done in Proposition 9.1 for λ large enough, without assuming that the action of G on X is cocompact.

Exponential growth rate of lacunary subsets
The goal of this section is to compare the exponential growth rates ofḠ * Z 2 andḠ ρ * Z 2 . We will use the notations (and obvious extensions) previously introduced without further notice.
Given σ > 0, there exists r σ > 0 such that (3.1) An explicit value for r σ can be obtained from a (naive) packing argument based on the following observation: every point ofX is at distance at most ∆ from a point of theḠ-orbit ofŌ and so at distance at most ∆ + ρ from some pointγ(Ō), whereγ ∈Ḡ ρ . Here, ∆ = diam(X/Ḡ) = diam(X/G) andŌ represents the projection of O toX.

Classical results about Gromov hyperbolic spaces
In this section, we recall the definition of Gromov hyperbolic spaces and present some well-known results. Classical references on the subject include [Gr87, GH90, CDP90].
for every x, y, z, w ∈ X. Equivalently, a metric space X is δ-hyperbolic if |xy| + |zw| ≥ max{|xz| + |yw|, |yz| + |xw|} + 2δ (4.2) for every x, y, z, w ∈ X. A metric space is Gromov hyperbolic if it is δ-hyperbolic for some δ ≥ 0. A finitely generated group is word hyperbolic in the sense of Gromov if its Cayley graph with respect to some (or equivalently any) finite generating set is Gromov hyperbolic.
Example 4.2. Gromov hyperbolic spaces include complete simply connected Riemannian manifolds of sectional curvature bounded away from zero and their convex subsets, metric trees and more generally CAT(−1) spaces.
Remark 4.3. In this definition, the metric space X is not required to be geodesic. However, from [BS00], every δ-hyperbolic metric space isometrically embeds into a complete geodesic δ-hyperbolic metric space.
Without loss of generality, we will assume in the sequel that X is a complete geodesic δ-hyperbolic metric spaces.
In this lemma, we also denoted by | · | the metric on T . We will refer to the map Φ as the approximation map of the geodesic triangle ∆.
Remark 4.5. This result implies the Rips condition: each side of a geodesic triangle of X is contained in the 4δ-neighborhood of the union of the other two sides.
Remark 4.6. Given x, y, z ∈ X, let Φ be the approximation map of a geodesic triangle ∆ = ∆(x, y, z) with vertices x, y and z. From Lemma 4.4.(1), the Gromov product (x|y) z is equal to the distance between Φ(z) and the center of the tripode T .
The following lemma is a simple version of the Local-to-Global theorem, cf. [Gr87, GH90, CDP90].
The following simple fact will be useful in the sequel.
Lemma 4.8. Let x ∈ X. Consider the projection p of x to a given geodesic line or geodesic segment τ (the projection may not be unique). Then, for every q ∈ τ , (x|q) p ≤ 4δ. In particular, |xq| ≥ |xp| + |pq| − 8δ.
The following classical result follows from the previous lemma. Proof. Let p be the projection to [x, y] of a point z in γ. From Lemma 4.8, we have the following two estimates Since p lies between x and y, we have |xy| = |xp| + |py|. Hence, Substituting this inequality into the upper bound (4.3), we obtain d(z, [x, y]) = |zp| ≤ 2 + 8δ.

Traveling along hyperbolic orbits
In this section, we introduce some definitions, notations and constructions which will be used throughout this article.
The following set of definitions and properties can be found in [Gr87, GH90, CDP90].
Definition 5.1. Let X be a proper geodesic δ-hyperbolic metric space. The boundary at infinity of X, denoted by ∂X, is defined as the equivalence classes of geodesic rays of X, where two rays are equivalent if they are at finite Hausdorff distance. Similarly, it can be defined as the equivalence classes of sequences of points (x i ) i≥1 of X whose Gromov product (x i |x j ) p with respect to any point p goes to infinity. Here, two such sequences (x i ) and (y j ) are equivalent if (x i |y j ) p goes to infinity. Thus, ∂X can be considered as a space of limit points. The space X ∪∂X, endowed with the natural topology extending the initial topology on X, is compact and contains X as an open dense subset. Given two distinct points a and b in ∂X, there is a geodesic line τ in X joining a to b, that is, τ (−∞) = a and τ (∞) = b. Every isometry of X uniquely extends to a homeomorphism of X ∪ ∂X. An isometry α of X is hyperbolic if for some (or equivalently any) point x ∈ X, the map n → α n (x) is a quasi-isometric embedding of Z into X. Alternatively, an isometry of X is hyperbolic if and only if it is of infinite order. Every hyperbolic isometry of X has exactly two fixed points on ∂X. An axis of a hyperbolic isometry of X is a geodesic line of X joining the two fixed points of the isometry. The minimal displacement of an isometry α of X is defined as dis(α) = inf x∈X |xα(x)|.
The minimal displacement of a hyperbolic isometry of X is positive.
Example 5.2. Let G be a group acting properly and cocompactly by isometries on a proper geodesic δ-hyperbolic metric space X. We know that G is a word hyperbolic group in the sense of Gromov (see the paragraph preceding Theorem 1.3). The argument leading to this result also shows that an element of G is hyperbolic as an isometry of X if and only if it is hyperbolic as an isometry of the Cayley graph of G with respect to any finite generating set (and so if and only if it is of infinite order). The boundary at infinity of the word hyperbolic group G, defined as the boundary at infinity of its Cayley graph with respect to some (and so any) finite generating set, agrees with ∂X. The group G is non-elementary if and only if ∂X contains more than two points. In this case, G contains a hyperbolic element. More generally, every infinite subgroup N of a word hyperbolic group G in the sense of Gromov contains a hyperbolic element.
Let G be a finitely generated group acting by isometries on a proper geodesic δ-hyperbolic metric space X. Suppose that G contains a hyperbolic isometry ξ of X, cf. Example 5.2. By taking a large enough power of ξ if necessary, we can assume that the minimal displacement of ξ, denoted by L = dis(ξ), is at most 300δ, i.e., L ≥ 300δ. Fix an axis τ of ξ.
Lemma 5.4. The point O is at distance at most 28δ from the axis τ of ξ.
Proof. Let p and q be the projections of O and ξ(O) to τ (they may not be unique but we choose some). Let also q − be the projection of ξ −1 (q) to τ . From [GH90,Corollaire 7.3], the image ξ −1 (τ ), which is a geodesic line with the same endpoints at infinity as τ , is at distance at most 16δ from τ . Thus, From this relation and Lemma 4.8, we derive That is, Now, from Lemma 4.7, we have where the third inequality follows from (5.1) and the last one from (5.2). By definition of O, we obtain this choice of indices may not seem natural, but it allows us to consider fewer cases in forecoming arguments). As ||ξ|| ∞ ≥ 0.9L, we derive that dis(ξ k ) ≥ ||ξ k || ∞ = |k| ||ξ|| ∞ ≥ L for every k ∈ Z * \ {±1}. Thus, dis(ξ k ) ≥ L for every k ∈ Z * . That is, We also define p i as the projection of x i to τ (again, it may not be unique but we choose one). Since ξ is an isometry, the points x i of the ξ-orbit of x 1 attain the minimal displacement of ξ up to . Thus, from Lemma 5.4, we derive For every pair of distinct indices i, j ∈ Z * , we obtain If the indices i and j are adjacent, we actually have L ± (56δ + ).
(5.4) using the notation 1.5. Therefore, since L > 168δ + 3 , we deduce that the points p i lie in τ in the order induced by Z * .
Consider the Voronoi cells of the ξ-orbit (x i ) of O, namely Lemma 5.5. Given x ∈ D i , let p be a projection of x to τ . Then p strictly lies between the points p i − and p i + of {p j | j ∈ Z * , j = i} adjacent to p i . In particular, |p i p| ≤ L + 56δ + .
Proof. By contradiction, we assume that there is an index j ∈ Z * adjacent to i such that p j lies between p i and p, or is equal to p. From (5.4), we have From the triangular inequality and the bound (5.3), this estimate leads to On the other hand, from Lemma 4.8, we have Hence, with the help of (5.3) and the inequality L > 120δ + , we obtain Thus, x does not lie in D i , which is absurd. Hence the first part of the lemma. The second part of the lemma follows from (5.4).
Remark 5.6. Lemma 5.5 also shows that two domains D i and D j corresponding to non-adjacent indices are disjoint. Thus, the Voronoi cells are ordered by their indices.
Lemma 5.7. Let x and y be two points of X separated by D ±1 or D ±2 . Then Proof. Let p and q be the projections of x and y to τ . By assumption, there exist two indices i, j ∈ Z * such that x ∈ D i and y ∈ D j with D ±1 or D ±2 separating D i and D j . From Lemma 5.5, p ±1 or p ±2 separates p and q. This point between p and q will be denoted by r. From the bounds (5.4) and (5.3), we derive that |Or| ≤ 2L + 140δ + 2 . Hence |pq| = |pr| + |rq| ≥ |Op| + |Oq| − 2 |Or| ≥ |Op| + |Oq| − 2(2L + 140δ + 2 ).

Geometric properties of symmetric elements
Using the previous notations and constructions, we introduce a notion of symmetric element in G and establish some (almost) norm-preserving properties.
We define β ± ∈ G as follows We will think of β − as the symmetric element of β with respect to a symmetry line perpendicular to the direction of the ξ-shift.

Geometric properties of twisted products
In this section, we introduce the twisted product and show that the norm on G is almost multiplicative with respect to the twisted product.
Definition 7.1. The twisted product of two elements α, β in G is defined as Note that the twisted product is not associative. Furthermore, it has no unit and is not commutative.

Inserting hyperbolic elements
The proposition established in this section will play an important role in the proof of the injectivity of the nonexpanding map defined in Section 9.
Proof. The first point is clear since ξ −κ (O) ∈ D −κ . We only check the second one. Let j = j(β) be the index of β, cf. Definition 6.1. Suppose that j > 0. We observe that the image of O by ξ κ β = ξ κ β lies in D j+κ and that the index of ξ κ β is positive equal to j + κ. Thus, Suppose that j < 0. We observe that the image of O by ξ κ β = ξ κ−2j+1 β lies in D −j+2+κ and that the index of ξ κ β is positive equal to −j + 2 + κ.
We denote by ξ the normal subgroup of G generated by ξ.
Proof. Let γ = α i (ξ κ β i ) = α i ξ κ i (β i ) ε i where κ i = ±κ or −κ − 4 and ε i = ± according to the decomposition (8.1) of Lemma 8.1. The segments , and from O to γ(O) will be denoted by a i , c i , b i and c (these segments may not be unique, but we choose some). We denote also byĀ i andB i the projections of A i and B i to c (which again may not be unique). Recall that ||ξ|| ≤ L + δ with 300δ ≤ L and observe that for κ ≥ 5, Thus, by Lemma 4.9, the arc a i ∪c i ∪b i lies at distance at most ∆ + 1 2 ∆ − +8δ from the geodesic c with the same endpoints. In particular, we have Fact 1. The points O,Ā i andB i are aligned in this order along c.
Indeed, from (8.3), we have This implies that and so  Let us prove the first assertion. Arguing by contradiction, we can assume from Fact 1 that the points O,Ā 2 ,B 2 andĀ 1 are aligned in this order along c. Combined with (8.4) and (8.5), this implies that Now, consider the projection where X/ ξ is endowed with the quotient distance, still denoted by | · |. By definition, Continuing with (8.6) and since π(A 2 ) = π(B 2 ), we obtain . Hence a contradiction from our choice of κ. Similarly, we derive the second assertion of Fact 2.

A nonexpanding embedding
In this section, we finally establish the key proposition in the proof of the main theorem. Namely, we construct a nonexpanding map Φ :Ḡ * Z 2 → G and show that it is injective in restriction to some separated subsetḠ ρ * Z 2 .
Let G be a finitely generated group acting by isometries on a proper geodesic δ-hyperbolic metric space X. Let N be a normal subgroup of G. The quotient groupḠ = G/N , whose neutral element is denoted byē, acts by isometries on the quotient metric spaceX = X/G.
Suppose that N contains a hyperbolic isometry ξ of X, cf. Example 5.2. By taking a large enough power of ξ if necessary, we can assume that the minimal displacement L of ξ is at least 300δ.
For every γ ∈Ḡ, we fix once and for all a representative α in G which is ν-minimal modulo N . (To avoid burdening the arguments by epsilontics, we will actually assume that ν = 0.) This yields an embedding Φ :Ḡ → G.
From this contradiction, we deduce that ε = ε .

Conclusion
Consider a non-elementary group G acting properly and cocompactly by isometries on a proper geodesic δ-hyperbolic metric space with fixed origin O. Denote by ∆ the diameter of X/G. Let N be an infinite normal subgroup of G andḠ = G/N . We can assume that ω(Ḡ) is nonzero. Denote by L the maximal value between 300δ and the minimal norm of a hyperbolic element in N .
Observe that if ω(Ḡ) is close to ω(G), then L is large, that is, the norm of every hyperbolic element of N is large.