Éphane St

Sabourau

GROWTH OF QUOTIENTS OF GROUPS ACTING BY ISOMETRIES ON GROMOV HYPERBOLIC SPACES

Keywords: 2010 Mathematics Subject Classification. Primary 20F69; Secondary 20F67, 20E07, 53C23 Exponential growth rate, entropy, critical exponent, asymptotic group theory, Gromov hyperbolic spaces, normal subgroups, covers, growth tightness. 1

We show that every group G with no cyclic subgroup of finite index that acts 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 unifies and extends previous works of Arzhantseva-Lysenok and Sambusetti using a geometric approach.

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

ω(G, d) = lim sup R→+∞ log cardB G (R) R (1.1)
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 group Ḡ = G/N of G by a normal subgroup N inherits the quotient distance d given by the least distance between representatives. The distance d 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 quotient Ḡ 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 and Ḡ clearly agree.

The notion of growth tightness was first introduced by R. Grigorchuk and P. de la Harpe [START_REF] Grigorchuk | On problems related to growth, entropy, and spectrum in group theory[END_REF] for word metrics on finitely generated groups. In this context, A. Sambusetti [START_REF] Sambusetti | Growth tightness of free and amalgamated products[END_REF] 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 [START_REF] Arzhantseva | Growth tightness for word hyperbolic groups[END_REF] gave an affirmative answer to the question about growth tightness of word hyperbolic groups posed by R. Grigorchuk and P. de la Harpe [START_REF] Grigorchuk | On problems related to growth, entropy, and spectrum in group theory[END_REF]. 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. [START_REF] Grigorchuk | On problems related to growth, entropy, and spectrum in group theory[END_REF].

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 group Ḡ = G/N by a normal subgroup N ¡ G is the deck transformation group of the normal cover M = M /N of M . The quotient distance dg on Ḡ agrees with the distance d ḡ on Ḡ induced by the lift ḡ on M of the Riemannian metric g on M (here, we take for a basepoint on M any lift of x 0 ). Furthermore, the exponential growth rate of ( Ḡ, d ḡ) agrees with the one of the Riemannian cover ( M , ḡ) defined as

ω( M , ḡ) = lim R→+∞ log vol B ḡ(R) R
where B ḡ(R) is the ball of radius R in M centered at the basepoint (the limit exists since M is compact). In other words, we have ω( Ḡ, d ḡ) = ω( M , ḡ). Note that the exponential growth rates of Ḡ and M 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).

In [START_REF] Sambusetti | Asymptotic properties of coverings in negative curvature[END_REF], A. Sambusetti proved the following Riemannian analogue of G. Arzhantseva and I. Lysenok's result [START_REF] Arzhantseva | Growth tightness for word hyperbolic groups[END_REF]. [START_REF] Sambusetti | Asymptotic properties of coverings in negative curvature[END_REF]). Every Riemannian normal cover M of a closed negatively curved Riemannian manifold M , different from the universal cover M , satisfies ω( M ) < ω( M ).

Theorem 1.1 ([
As pointed out in [START_REF] Sambusetti | Asymptotic properties of coverings in negative curvature[END_REF], 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 [START_REF] Sambusetti | Growth tightness in group theory and Riemannian geometry[END_REF], 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, | 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. [START_REF] Coornaert | Mesures de Patterson-Sullivan sur le bord d'un espace hyperbolique au sens de Gromov[END_REF].

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 group Ḡ 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. [START_REF] Arzhantseva | Growth tightness for word hyperbolic groups[END_REF]. 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

• Riemannian metrics on M without curvature assumption (and even to Finsler metrics); • manifolds with a different topology type than M , such as the nonaspherical manifolds M #(S 1 × S n-1 ) or M #M 0 , where M 0 is any simply connected closed n-manifold different from S n ; • intermediate covers and more generally covering towers, whereas Theorem 1.1 only deals with the universal cover of M , see the following corollary for an illustration.

Corollary 1.4. Let M be a Riemannian normal cover of a closed Riemannian manifold M . Suppose that M is Gromov hyperbolic and that its boundary at infinity contains more than two points. Then every Riemannian normal cover

M → M → M with M = M satisfies ω( M ) < ω( M ).
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 [START_REF] Margulis | Discrete subgroups of semisimple Lie groups[END_REF]. 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 cover M of M (their exponential growth rate agrees with the exponential growth rate of M ).

In [DPPS11, §5.1], building upon constructions of [START_REF] Dal'bo | Séries de Poincaré des groupes géométriquement finis[END_REF], 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 for M = 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 and Ḡ can be arbitrarily small. For free groups, this follows from [START_REF] Shukhov | On the dependence of the growth exponent on the length of the defining relation[END_REF], see also [START_REF] Talambutsa | Attainability of the index of exponential growth in free products of cyclic groups[END_REF]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 [START_REF] Sambusetti | On minimal growth in group theory and Riemannian geometry[END_REF][START_REF] Sambusetti | Growth tightness of surface groups[END_REF][START_REF] Sambusetti | Growth tightness of free and amalgamated products[END_REF][START_REF] Sambusetti | Growth tightness of negatively curved manifolds[END_REF][START_REF] Sambusetti | Growth tightness in group theory and Riemannian geometry[END_REF][START_REF] Sambusetti | Asymptotic properties of coverings in negative curvature[END_REF][START_REF] Dal'bo | On the growth of quotients of Kleinian groups[END_REF]. 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.

Notation 1.5. Given A, B, C ≥ 0, it is convenient to write A B ± C for B -C ≤ A ≤ B + C.
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 group Ḡ = G/N by a normal subgroup N ¡G acts properly and cocompactly by isometries on the quotient metric space X = X/N . We will assume that ω( Ḡ) is nonzero, otherwise the main result clearly holds. This implies that X is unbounded. 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 on Ḡ from the quotient pseudo-distance d. 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".

For λ ≥ 0, consider the norm || • || λ on the free product Ḡ * Z 2 where

||γ 1 * 1 * • • • * γ m+1 || λ = m+1 i=1 ||γ i || Ḡ + mλ for every element γ 1 * 1 * • • • * γ m+1 ∈ Ḡ * Z 2 , with γ i ∈ Ḡ, in reduced form (that is, with m minimal). The norm || • || λ induces a left-invariant distance, denoted by d λ , on Ḡ * Z 2 . We will write ω( Ḡ * Z 2 , λ) for the exponential growth rate of ( Ḡ * Z 2 , d λ ). Clearly, ω( Ḡ * Z 2 , λ + λ ) ≤ ω( Ḡ * Z 2 , λ) for every λ, λ ≥ 0. A direct combinatorial computation [Sa02b, Proposition 2.3] shows that ω( Ḡ * Z 2 , λ) > ω( Ḡ).
More precisely, the following estimate holds Proposition 2.1 (see Proposition 2.3 in [START_REF] Sambusetti | Growth tightness of free and amalgamated products[END_REF]).

For every λ > diam(X/G), we have

ω( Ḡ * Z 2 , λ) ≥ ω( Ḡ) + 1 4λ log(1 + e -λω( Ḡ) ).
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

( Ḡ * Z 2 , || • || λ ) → (G, || • || G ), we could claim that the R-ball B Ḡ * Z 2 ,λ (R) of Ḡ * 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. Let Ḡρ be a subset of Ḡ containing the neutral element ē ∈ Ḡ such that the elements of Ḡρ are at distance greater than ρ from each other and every element of Ḡ is at distance at most ρ from an element of Ḡρ .

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 set Ḡρ is formed of all words of even length.

Consider the subset Ḡρ

* Z 2 of Ḡ * Z 2 formed of the elements γ 1 * 1 * • • • * γ m+1 for m ∈ N with γ i ∈ Ḡρ .
Suppose we can construct an injective 1-Lipschitz map

Φ : ( Ḡρ * Z 2 , || • || λ ) → (G, || • || G ) (2.1)
for λ large enough. Then, as previously, the R-ball B Ḡρ * Z2,λ (R) injects into an R-ball of G and so ω(

Ḡρ * Z 2 , λ) ≤ ω(G).
(2.2) To derive the main theorem, we simply need to compare ω( Ḡρ * Z 2 , λ) with ω( Ḡ * Z 2 , λ). This is done in the next section, cf. Proposition 3.1, where we show that

ω( Ḡ * Z 2 , 2λ + λ ) ≤ ω( Ḡρ * Z 2 , λ) (2.3)
for λ large enough. The combination of Proposition 2.1, applied to 2λ + λ , with (2.3) and (2.2) allows us to conclude.

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 of Ḡ * Z 2 and Ḡρ * 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 cardB Ḡρ (r σ ) ≥ cardB Ḡ(σ).

(3.1)

An explicit value for r σ can be obtained from a (naive) packing argument based on the following observation: every point of X is at distance at most ∆ from a point of the Ḡ-orbit of Ō and so at distance at most ∆ + ρ from some point γ( Ō), where γ ∈ Ḡρ . Here, ∆ = diam( X/ Ḡ) = diam(X/G) and Ō represents the projection of O to X.

More precisely, let > 0. As X is unbounded, we can take cardB Ḡ(σ) points (x i ) on a minimizing ray of X of length = 2(∆ + ρ) cardB Ḡ(σ) + based at Ō with d(x i , xj ) > 2(∆ + ρ) for i = j. From the previous observation, every point xi is at distance at most ∆ + ρ from some γi ( Ō), where γi ∈ Ḡρ . By construction, the elements γi are disjoint and lie in B Ḡρ ( + ∆ + ρ). Thus, the bound (3.1) holds with

r σ = 3(∆ + ρ) cardB Ḡ(σ).
By applying the previous observation to a point of X at distance ∆+ρ+ from Ō, we can also show that there exists θ ∈ Ḡρ different from ē with || θ|| Ḡ ≤ 2(∆ + ρ).

Proposition 3.1. We have

ω( Ḡ * Z 2 , λ + λ ) ≤ ω( Ḡρ * Z 2 , λ
2 ) where λ ≥ 0 and λ ≥ R ρ = 6(∆ + ρ) cardB Ḡ(3(∆ + ρ)).

Proof. Let λ ≥ r σ + σ where σ = 3(∆ + ρ). For every γ ∈ Ḡ, we define γ ∈ Ḡρ as follows. If γ ∈ B Ḡ(ρ), then γ = θ. Otherwise, we choose for γ an element of Ḡρ at distance at most ρ from γ. By construction, we have

d(γ,γ) ≤ 2(∆ + ρ) + ρ ≤ 3(∆ + ρ) = σ. Now, we consider the map ϕ : Ḡ * Z 2 → Ḡρ * Z 2 defined as ϕ(γ 1 * 1 * • • • * γ m+1 ) = γ1 * 1 * • • • * γm+1 .
As no γi agrees with ē (this is the reason for introducing θ), the product γ1

* 1 * • • • * γm+1 is in reduced form. It follows from the bound d(γ,γ) ≤ σ that ||γ 1 * 1 * • • • * γm+1 || 2λ+λ -σ = m+1 i=1 ||γ i || Ḡ + m(2λ + λ -σ) ≤ m+1 i=1 ||γ i || Ḡ + (m + 1)σ + m(2λ + λ -σ) ≤ ||γ 1 * 1 * • • • * γ m+1 || 2λ+λ + σ.
Therefore, the map ϕ sends

B Ḡ * Z 2 ,2λ+λ (R) to B Ḡρ * Z2,2λ+λ -σ (R + σ).
Furthermore, the bound d(γ,γ) ≤ σ also implies that every element γ1 *

1 * • • • * γm+1 ∈ Ḡρ * Z 2 in reduced form has at most cardB Ḡ(γ 1 , σ) × • • • × cardB Ḡ(γ m+1 , σ) = (cardB Ḡ(σ)) m+1
preimages by ϕ.

Hence, the cardinal of B Ḡ * Z 2 ,2λ+λ (R) is bounded by the following sum

∞ m=0 card{γ 1 * 1 * • • • * γm+1 ∈ B Ḡρ * Z2,2λ+λ -σ (R + σ)} (cardB Ḡ(σ)) m+1 .
Now, since cardB Ḡ(σ) ≤ cardB Ḡρ (r σ ), each term of this sum is bounded by the number of elements

γ1 * 1 * • • • * γm+1 * 1 * c1 * 1 * • • • * cm+1 ∈ Ḡρ * Z 2 such that m+1 i=1 ||γ i || Ḡ + m(2λ + λ -σ) ≤ R + σ
and ci ∈ B Ḡρ (r σ ). As all these elements satisfy m+1 i=1

||γ i || Ḡ + m+1 i=1 ||c i || Ḡ + m(2λ + λ -σ -r σ ) ≤ R + σ + r σ and so m+1 i=1 ||γ i || Ḡ + m+1 i=1 ||c i || Ḡ + (2m + 1) (λ + λ -σ -r σ 2 ) ≤ R + λ + λ since λ ≥ r σ + σ, we derive that cardB Ḡ * Z 2 ,2λ+λ (R) ≤ cardB Ḡρ * Z2,λ+ λ -σ-rσ 2 (R + λ + λ ).
Therefore,

ω( Ḡ * Z 2 , 2λ + λ ) ≤ ω( Ḡρ * Z 2 , λ + λ -σ -r σ 2 ) ≤ ω( Ḡρ * Z 2 , λ)
from the nonincreasing property of ω( Ḡρ * Z 2 , •).

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]. Definition 4.1. Let X = (X, | • |) be a metric space. The Gromov product of x, y ∈ X with respect to a basepoint w ∈ X is defined as for every x, y, z, w ∈ X.

(x|y) w = 1 2 (|xw| + |yw| -|xy|). ( 4
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 [START_REF] Bonk | Embeddings of Gromov hyperbolic spaces[END_REF], 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.

The following result about approximation maps can be found in [START_REF] Gromov | Hyperbolic groups[END_REF], [GH90, Ch. 2, §3] and [START_REF] Coornaert | Géométrie et théorie des groupes. Les groupes hyperboliques de Gromov[END_REF].

Lemma 4.4. Given a geodesic triangle ∆ in X, there exists a map Φ : ∆ → T to a possibly degenerated tripode T (i.e., a metric tree with at most three leaves) such that

(1) the restriction of Φ to each edge of ∆ is an isometry, (2) for every x, y ∈ X,

|xy| -4δ ≤ |Φ(x)Φ(y)| ≤ |xy|.
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].

Lemma 4.7. Let x, y, p, q ∈ X such that (x|q) p ≤ 4δ and (y|p) q ≤ 4δ. Then Combined with the previous two estimates, we obtain the desired lower bound for |xy|.

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.

Proof. Consider an approximation map Φ : ∆ → T of a geodesic triangle ∆ = ∆(x, p, q) with vertices x, p and q. The center of the tripode T has three preimages by Φ, one in each segment [x, p], [p, q] and [x, q]. Let

x

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.

Definition 5.3. The stable norm of the fixed hyperbolic element ξ is defined as the limit of the subadditive sequence ||ξ k || k . It is denoted by ||ξ|| ∞ . From [CDP90, §10, Proposition 6.4], the stable norm of ξ is positive and satisfies

||ξ|| ∞ ≤ L = dis(ξ) ≤ ||ξ|| ∞ + 16δ.
Observe that ||ξ|| ∞ ≥ 0.9L as L ≥ 300δ.

Given ∈ (0, δ), let O ∈ X be an origin with |Oξ(O)| ≤ L + .

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,

|ξ -1 (q)q -| ≤ 16δ. (5.1)
From this relation and Lemma 4.8, we derive

|Op| + |pq -| -8δ ≤ |Oq -| ≤ |Oξ -1 (q)| + 16δ. That is, |pq -| ≤ |ξ(O)q| -|Op| + 24δ. (5.2)
Now, from Lemma 4.7, we have

|Oξ(O)| ≥ |Op| + |pq| + |qξ(O)| -14δ ≥ |Op| + |ξ -1 (q)q| -|ξ -1 (q)p| + |qξ(O)| -14δ ≥ |Op| + |qξ(q)| -|pq -| -16δ + |qξ(O)| -14δ ≥ |qξ(q)| + 2|Op| -54δ
where the third inequality follows from (5.1) and the last one from (5.2). By definition of O, we obtain

|Op| ≤ 27δ + 2 < 28δ. Let x 1 = O. For i ∈ Z * = Z \ {0}, we denote by x i the point ξ i-1 (x 1 ) if i > 0 and ξ i (x 1 ) if i < 0 (

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, |x i x j | ≥ L
for every i, j ∈ Z * with i = j. 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 |x i p i | ≤ 28δ.

(5.3)

For every pair of distinct indices i, j ∈ Z * , we obtain

|p i p j | ≥ |x i x j | -56δ ≥ L -(56δ + ).
If the indices i and j are adjacent, we actually have

|p i p j | |x i x j | ± 56δ 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

D i = {x ∈ X | |xx i | ≤ |xx j | for every j ∈ Z * }. Note that ξ(D i ) = D i+1 if i ∈ Z * \ {-1} and ξ(D -1 ) = D 1 .
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

|pp j | = |pp i | -|p i p j | ≤ |pp i | -L + 56δ + .
From the triangular inequality and the bound (5.3), this estimate leads to

|xx j | ≤ |xp| + |pp j | + |p j x j | ≤ |xp| + |pp i | -L + 56δ + + 28δ.
On the other hand, from Lemma 4.8, we have

|xp| + |pp i | ≤ |xp i | + 8δ ≤ |xx i | + |x i p i | + 8δ.
Hence, with the help of (5.3) and the inequality L > 120δ + , we obtain

|xx j | ≤ |xx i | + 28δ + 8δ -L + 56δ + + 28δ < |xx i |.
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 |xy| ≥ |Ox| + |Oy| -4L -280δ -4 .
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. Definition 6.1. Let β ∈ G. Denote by j(β) the smallest index j ∈ Z * such that β(O) ∈ D j . We say that β is positive if j(β) > 0 and negative if j(β) < 0.

We define β ± ∈ G as follows

β + = β and β -= ξ -2j-1 β if β is positive ξ -2j+1 β if β is negative
where j = j(β).

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.

Consider the following norm on G defined for every β ∈ G as

||β|| = |Oβ(O)|. Proposition 6.2. Let β, β 1 , β 2 ∈ G and j = j(β). (1) β -(O) lies in D -j-2 if j > 0 and in D -j+2 if j < 0.
In particular, β + and β -have opposite signs.

(2) If (β 1 ) -= (β 2 ) -then β 1 = β 2 . (3) We have ||β -|| ||β|| ± ∆ - where ∆ -= ∆ -(L, δ, ) = 8L + 464δ + 8 ≤ 10L.
Proof. The point (1) is obvious by construction of β -.

For the second point, if (β 1 ) -= (β 2 ) -then β 1 and β 2 have the same sign from the second assertion of (1). We derive from the definition of β -that

ξ k β 1 = ξ k β 2 for k = -2j ± 1. Hence β 1 = β 2 .
For the last point, let x = β(O) and

x -= β -(O) -= ξ -2j∓1 β(O)
, where the exponent -2j ∓ 1 depends on the sign of β, see Definition 6.1. Fix k = -j ∓ 2. By assumption, x ∈ D j and from the point (1), x -∈ D k . Thus,

|xx j | = |x -x k |.
Let p and p -be the projections of x and x -to τ . From Lemma 5.5 and the bound (5.3), both |x j p| and |x k p -| are bounded from above by L + 84δ + . Combined with

|xp| |xx j | ± |x j p| |x -p -| |x -x k | ± |x k p -| and the relation |xx j | = |x -x k |, we obtain |xp| |x -p -| ± (2L + 168δ + 2 ). (6.1)
On the other hand, from (5.3) and the previous bound on |x j p|, we derive

|p 1 p| |x 1 x j | ± (|p 1 x 1 | + |x j p|) |x 1 x j | ± (L + 112δ + ) (6.2) Similarly, |p 1 p -| |x 1 x k | ± (L + 112δ + ). (6.3) Now, since |x 1 x j | = |x -j+2 x 1 | |x k x 1 | ± 4(L + )
we obtain from (6.2) and (6.3) that

|p 1 p| |p 1 p -| ± (6L + 224δ + 6 ). (6.4)
Now, from Lemma 4.8, we have

|xp 1 | |xp| + |pp 1 | ± 8δ |x -p 1 | |x -p -| + |p -p 1 | ± 8δ
Combined with (6.1) and (6.4), we obtain

|xp 1 | |x -p 1 | ± (8L + 408δ + 8 ).
The result follows from the bound |Op 1 | ≤ 28δ derived in (5.3).

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

α β = αβ + if D 1 or D -1 separates α -1 (O) and β(O) αβ -otherwise
Note that the twisted product is not associative. Furthermore, it has no unit and is not commutative. (2). Switching β and β if necessary, we can assume that αβ = α(β ) -, and so β = (β ) -. From Proposition 6.2.(1), this implies that β and β have opposite signs, which is absurd.

For the last point. We have

||α β|| = |O αβ ε (O)| = |α -1 (O)β ε (O)|.
Using the point (1), we can apply Lemma 5.7 to derive

|Oα -1 (O)| + |Oβ ε (O)| -4L -280δ -4 ≤ ||α β|| ≤ |Oα -1 (O)| + |Oβ ε (O)|.
That is,

||α β|| ||α|| + ||β ε || ± (4L + 280δ + 4 ).
Hence, by Proposition 6.2, we obtain ||α β|| ||α|| + ||β|| ± (12L + 744δ + 12 ).

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.

Lemma 8.1. Let β ∈ G and κ ≥ 4.

(1)

(ξ κ β) + = ξ κ β + if β is positive ξ κ β -if β is negative
In particular, (ξ κ β) + is positive.

(2)

(ξ κ β) -= ξ -κ β -if β is positive ξ -κ-4 β + if β is negative
In particular, (ξ κ β) -is negative. Therefore, for every α ∈ G, we have

α (ξ κ β) = α ξ κ * β ε (8.1)
with κ * = ±κ or -κ -4, and ε = ±.

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, (ξ κ β) -= ξ -2(j+κ)-1 ξ κ β = ξ -κ β -.

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. Thus, (ξ κ β) -= ξ -2(-j+2+κ)-1 ξ κ-2j+1 β = ξ -κ-4 β.

||ξ|| ∞ ≤ L = dis(ξ) ≤ ||ξ|| ∞ + 16δ. Proposition 8.3. Let α 1 , α 2 , β 1 , β 2 ∈ G such that α i is η-minimal mod- ulo ξ for some η ≥ 0. Let κ ≥ 5 be an integer such that κ > (6∆ + 3∆ -+ 48δ + η)/||ξ|| ∞ . If α 1 (ξ κ β 1 ) = α 2 (ξ κ β 2 ), then d(α 1 , α 2 ) ≤ 2κ L + 4(∆ + 1 2 ∆ -+ 8δ).
Remark 8.4. Recall that L ≥ 300δ, ∆ -≤ 10L and ∆ ≤ 15L. Hence, when η = ∆ -, we can take κ = 140.

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 from O to

A i = α i (O), from A i to B i = α i (ξ κ i (O)), from B i to γ(O)
, 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 by Āi and Bi 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,

κ ||ξ|| ∞ ≤ |κ i | ||ξ|| ∞ ≤ |A i B i | = ||ξ κ i || ≤ (κ + 4)||ξ|| ≤ 2κ L. (8.2)
From Propositions 6.2.(3) and 7.2.(3), we derive

length(a i ∪ c i ∪ b i ) = ||α i || + ||ξ κ i || + ||(β i ) ε i || ||α i (ξ κ β i )|| ± (2∆ + ∆ -) length(c) ± (2∆ + ∆ -). (8.3)
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

|A i Āi | ≤ ∆ + 1 2 ∆ -+ 8δ (8.4) |B i Bi | ≤ ∆ + 1 2 ∆ -+ 8δ (8.5)
Fact 1. The points O, Āi and Bi are aligned in this order along c.

Indeed, from (8.3), we have

|OA i | + |A i B i | + |B i γ(O)| -2∆ -∆ -≤ |Oγ(O)| ≤ |OB i | + |B i γ(O)|.
This implies that

|OB i | |OA i | + |A i B i | ± (2∆ + ∆ -)
and so Let us prove the first assertion. Arguing by contradiction, we can assume from Fact 1 that the points O, Ā2 , B2 and Ā1 are aligned in this order along c. Combined with (8.4) and (8.5), this implies that

|O Bi | ≥ |O Āi | + κ ||ξ|| ∞ -(2∆ + ∆ -) -2(∆ + 1 2 ∆ -+ 8δ
|OA 1 | |O Ā1 | ± (∆ + 1 2 ∆ -+ 8δ) |O Ā2 | + | Ā2 B2 | + | B2 Ā1 | ± (∆ + 1 2 ∆ -+ 8δ) |OA 2 | + |A 2 B 2 | + |B 2 A 1 | ± 6(∆ + 1 2 ∆ -+ 8δ) (8.6)
Now, consider the projection

π : X → X/ ξ ,
where X/ ξ is endowed with the quotient distance, still denoted by | • |. By definition, A i = α i (O) and

B i = α i (ξ κ i (O)) = (α i ξ κ i α -1 i )(α i (O)). Hence, π(A i ) = π(B i ) since α i ξ κ i α -1 i ∈ ξ . Note also that |OA i | |π(O)π(A i )| ± η since α i is η-minimal modulo ξ .
Continuing with (8.6) and since π(A 2 ) = π(B 2 ), we obtain

|OA 1 | ≥ |π(O)π(A 2 )| + κ ||ξ|| ∞ + |π(B 2 )π(A 1 )| -6(∆ + 1 2 ∆ -+ 8δ) ≥ |π(O)π(A 1 )| + κ ||ξ|| ∞ -6(∆ + 1 2 ∆ -+ 8δ) ≥ |OA 1 | -η + κ ||ξ|| ∞ -6(∆ + 1 2 ∆ -+ 8δ
). Hence a contradiction from our choice of κ. Similarly, we derive the second assertion of Fact 2.

From the order of the points Āi and Bi on c given by Facts 1 and 2, we deduce by (8.2), (8.4) and (8.5) that

| Ā1 Ā2 | ≤ max{| Ā1 B1 |, | Ā2 B2 |} ≤ 2κ L + 2(∆ + 1 2 ∆ -+ 8δ). Therefore, d(α 1 , α 2 ) = |A 1 A 2 | ≤ 2κ L + 4(∆ + 1 2 ∆ -+ 8δ).

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 subset Ḡρ * 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 group Ḡ = G/N , whose neutral element is denoted by ē, acts by isometries on the quotient metric space X = 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.

Given 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. For ρ > 0, consider a maximal system of disjoint (closed) balls of radius ρ/2 in Ḡ containing the ball of radius ρ/2 centered at ē. By maximality, the centers of these balls form a subset Ḡρ of Ḡ such that (1) the elements of Ḡρ are at distance greater than ρ from each other;

(2) every element of Ḡ is at distance at most ρ from an element of Ḡρ . Note that ||γ|| Ḡ > ρ for every γ ∈ Ḡρ with γ = ē.

Consider the subset Ḡρ * Z 2 of Ḡ * Z 2 formed of the elements

γ 1 * 1 * • • • * γ m+1
where m ∈ N and γ i ∈ Ḡρ . Note that Ḡρ * Z 2 is not a group, as Ḡρ itself is not a group. (9.1)

Let α 1 = Φ(γ 1 ) and ε be the sign of Φ(γ 2 * 1 * • • • * γ m+1 ), unless γ 1 = ē1 in which case ε = +. Similar definitions hold for α 1 and ε by replacing γ i with γ i . By Proposition 6.2.(3), for every α ∈ G which is minimal modulo N , and so modulo ξ , its symmetric α -is η-minimal modulo ξ with η = ∆ -. Thus, from Proposition 8.3, we have d((α 1 ) ε , (α 1 ) ε ) ≤ ρ.

By definition of the quotient distance, d(γ 1 , γ 1 ) ≤ d(ξ i α 1 , ξ i α 1 ) for every i, i ∈ Z. Hence, d(γ 1 , γ 1 ) ≤ d((α 1 ) ε , (α 1 ) ε ) ≤ ρ.

Since γ 1 and γ 1 lie in a ρ-separated set, this shows that γ 1 = γ 1 and so α 1 = α 1 .

Suppose ε = ε . As ε and ε play symmetric roles, we can assume that ε = -. In this case, α = α 1 is different from ē and d(α, α -) ≤ ρ. Now, as α -1 α = α -1 α -, we derive from Proposition 7.2.(3) that

d(α, α -) = ||α -1 α -|| = ||α -1 α|| ≥ 2||α|| -∆ ≥ 2ρ -∆ > ρ.
From this contradiction, we deduce that ε = ε .

Substituting this equality into (9.1), we derive

α ε (ξ κ β) = α ε (ξ κ β )
where β = Φ(γ 2 * 1 * • • • * γ m+1 ) and with a similar definition for β by replacing γ i with γ i . As both ξ κ β and ξ κ β are positive, cf. Lemma 8.1.(1), we obtain from Proposition 7.2.(2) that ξ κ β = ξ κ β . Since β and β are of the same sign (i.e., ε = ε ), we similarly derive that β = β . We conclude by induction that m = m and γ i = γ i .

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 and Ḡ = 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 .

  • |) with origin O. The distance on X induces a left-invariant pseudo-metric d on G given by d(α, β) = |α(O)β(O)| (1.3) 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.

  Fix an origin O ∈ X. The left-invariant pseudo-distance d, cf. (1.3), induces a semi-norm || • || G on G given by ||α|| G = d(e, α) 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||

  |xy| ≥ |xp| + |pq| + |qy| -14δ. Proof. From the definition of the Gromov product, cf. (4.1), the bounds (x|q) p ≤ 4δ and (y|p) q ≤ 4δ yield the following two estimates |xq| ≥ |xp| + |pq| -8δ |yp| ≥ |yq| + |qp| -8δ. Now, from (4.2), we have |xy| + |pq| ≥ max{|xp| + |yq|, |yp| + |xq|} + 2δ ≥ |xq| + |yp| + 2δ.

  Recall that ||α|| = |Oα(O)| for every α ∈ G. Proposition 7.2. Let α, β, β ∈ G. Then (1) If α β = αβ ε with ε = ± then D ±1 or D ±2 separates α -1 (O) and β ε (O). (2) If α β = α β with β and β of the same sign, then β = β .

( 3 )

 3 We have ||α β|| ||α|| + ||β|| ± ∆ where ∆ = ∆ (L, δ, ) = 12L + 744δ + 12 ≤ 15L. Proof. For the first point. If ε = +, then the result is clear by definition of the twisted product. Namely, D ±1 separates α -1 (O) and β ε (O). If ε = -, then D 1 and D -1 do not separate α -1 (O) and β(O). Assume that β is positive, that is, β(O) ∈ D j with j > 0 (the other case is similar). This implies that α -1 (O) lies in a domain D i with i ≥ -1, otherwise D -1 would separate the two points. Therefore, D -2 separates α -1 (O) and β -(O) since β -(O) lies in D -j-2 from Proposition 6.2. For the second point, if αβ = αβ then β = β . Similarly, if α(β) -= α(β ) -then β = β from Proposition 6.2.

Definition 8. 2 .

 2 An element α ∈ G is η-minimal modulo a normal subgroup N ¡ G, where η ≥ 0, if for every α ∈ αN , ||α|| ≤ ||α || + η. We denote by ξ the normal subgroup of G generated by ξ. Recall that d(α, β) = |α(O)β(O)| for every α, β ∈ G and that the stable norm of ξ introduced in Definition 5.3 satisfies

  ) using (8.4), (8.5) and (8.2). By our choice of κ, we have|O Bi | ≥ |O Āi |.Fact 2. The points O, Ā1 and B2 are aligned in this order along c. Similarly, the points O, Ā2 and B1 are aligned in this order along c.

  ∈ (0, δ), let O ∈ X be an origin with |Oξ(O)| ≤ L + . Denote by || • || and || • || Ḡ the norms induced on G and Ḡ by the distance d, cf. (1.3), and the quotient distance d. We also need to fix κ = 140, cf. Remark 8.4.

  We extend this embedding to a map Φ : Ḡ * Z 2 → G by induction on m with the relationΦ(γ 1 * 1 * • • • * γ m+1 ) = Φ(γ 1 ) ε (ξ κ Φ(γ 2 * 1 * • • • * γ m+1 ))whereγ 1 * 1 * • • • * γ m+1 is in reduced form and ε is the sign of Φ(γ 2 * 1 * • • • * γ m+1 ), unless γ 1 = ē, in which case ε = +.For λ ≥ 0, consider the norm || • || λ on the group Ḡ * Z 2 defined as||γ 1 * 1 * • • • * γ m+1 || λ = m+1 i=1 ||γ i || Ḡ + mλ.Let α i = Φ(γ i ). From Propositions 6.2.(3) and 7.2.(3), we derive||Φ(γ 1 * 1 * • • • * γ m+1 )|| ≤ ||α 1 || + ||ξ κ || + ||Φ(γ 2 * 1 * • • • * γ m+1 )|| + 2∆ + ∆ - ≤ m+1 i=1 ||α i || + m(κL + 2∆ + ∆ -)Thus, since ||α i || = ||γ i || Ḡ, the map Φ does not increase the "norms" if λ ≥ κL + 2∆ + ∆ -.

Proposition 9. 1 .

 1 Let λ ≥ κL+2∆ +∆ -and ρ ≥ 2κL+4(∆ + 1 2 ∆ -+8δ). Then the nonexpanding map Φ : ( Ḡρ * Z 2 , || • || λ ) → (G, || • ||) is injective.Remark 9.2. We can take λ = 200L and ρ = 500L.Proof. Assume Φ(γ 1 * 1 * • • • * γ m+1 ) = Φ(γ 1 * 1 * • • • * γ m +1 ).

Combining Propositions 2.1, 3.1 and 9.1, we obtain

where λ = 200L, ρ = 500L and λ = 2λ + 6(∆ + ρ) cardB Ḡ(3(∆ + ρ)). One could also replace cardB Ḡ(3(∆ + ρ)) with cardB G (3(∆ + ρ)), since the latter cardinal is at least as large as the former.

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.