Hyperbolic and cubical rigidities of Thompson’s group V

1 Introduction

A major theme in geometric group theory is to make a given group act on a metric space which belongs to a specific class 𝒞 in order to deduce some information about it. However, not every group is sensitive to a given class of spaces, meaning that every isometric action of a fixed group on any one of these spaces may turn out to be trivial in some sense. Nevertheless, although the machinery of group actions on spaces of 𝒞 cannot be applied, it turns out that the non-existence of good actions provides interesting information as well. Roughly speaking, it implies some rigidity phenomena.

The first occurrence of such an idea was Serre’s Property (FA). A group satisfies Property (FA) if every isometric action on a simplicial tree fixes a point. We refer to [45, Section 6] for more information about this property. For instance, Property (FA) imposes restrictions on how to embed a given group into another (see for instance [24] in the context of 3-manifolds), and more generally on the possible homomorphisms between them (see for instance [21, Corollary 4.37] in the context of relatively hyperbolic groups). Also, such a rigidity has been applied in [33] to determine when the fundamental groups of two graphs of groups whose vertex-groups satisfy Property (FA) are isomorphic.

Another famous fixed-point property is Kazhdan’s Property (T). Usually, Property (T) is defined using representation theory, but alternatively, one can say that a (discrete) group satisfies Property (T) if every affine isometric action on a Hilbert space has a global fixed point, or equivalently if every isometric action on a median space has bounded orbits. See [2] and [13] for more information. Property (T) for a group imposes, for instance, strong restrictions on the possible homomorphisms starting from that group (for a geometric realisation of this idea, see for example [42], whose main construction has been very inspiring in other contexts), and plays a fundamental role in several rigidity statements, including the famous Margulis’ superrigidity. We refer to [2], and in particular to its introduction, for more information about Property (T).

In this article, we are mainly interested in the class of Gromov-hyperbolic spaces. We say that a group is hyperbolically elementary if every isometric action on a hyperbolic space either fixes a point at infinity, or stabilises a pair of points at infinity, or has bounded orbits. Once again, such a property imposes restrictions on the possible homomorphisms between two groups. For instance, it is proved in [32] that higher rank lattices are hyperbolically elementary, from which it is deduced that any morphism from a higher rank lattice to the mapping class group of a closed surface with punctures must have finite image (a statement originally due to Farb, Kaimanovich and Masur). It is worth noticing that a hyperbolically elementary group either satisfies Serre’s Property (FA) or surjects onto ℤ or D∞, so that being hyperbolically elementary is essentially a generalisation of Property (FA), which is not necessarily much harder to prove, see for instance [20] about branch groups.

We emphasise the fact that it is not reasonable to remove the possibility of fixing a point at infinity from the definition of hyperbolic elementarity. Indeed, any infinite group admits a proper and parabolic action on a hyperbolic space; see for instance the classical construction explained in [35, Section 4]. However, being hyperbolically elementary does not mean that any isometric action on a hyperbolic space is completely trivial, since the definition does not rule out lineal actions (i.e., actions on a quasi-line) nor quasi-parabolic actions (i.e., actions with loxodromic isometries all sharing a point at infinity). And these actions may provide interesting information on a group. For instance, admitting lineal actions is related to the existence of quasimorphisms; and admitting a quasi-parabolic action implies the existence of free sub-semigroups, so that the group must have exponential growth.

The first main objective of our article is to prove a general criterion leading to some hyperbolic rigidity. More precisely, we have the following theorem.

Correction added on 17 January 2019 after online publication: The second item has been modified (producing a more general statement). The previous statement and its proof were correct, but the new formulation allowed us to correct a mistake in the proof of Theorem 4.6 below.

Our main motivation in proving this criterion is to show that Thompson’s group V is hyperbolically elementary.

The groups F, T and V were defined by Richard Thompson in 1965. Historically, Thompson’s groups T and V are the first explicit examples of finitely presented simple groups. Thompson’s groups were also used in [39] to construct finitely presented groups with unsolvable word problems, and in [47] to show that a finitely generated group has a solvable word problem if and only if it can be embedded into a finitely generated simple subgroup of a finitely presented group. We refer to [10] for a general introduction to these three groups. Since then, plenty of articles have been dedicated to Thompson’s groups, and they have been the source of inspiration for the introduction of many classes of groups, now referred to as Thompson-like groups; see for instance [34, 7, 46, 6, 25, 5, 3]. Nevertheless, Thompson’s groups remain mysterious, and many questions are still open. For instance, it is a major open question to know whether F is amenable, and the structure of subgroups of V is still essentially unknown [8].

Our initial motivation in proving Theorem 1.2 came from another fixed-point property, in the class of CAT(0) cube complexes. A group G

1. satisfies Property (FWn), for some n≥0, if every isometric action of G on an n-dimensional CAT(0) cube complex has a global fixed point,

2. satisfies Property (FW∞) if it satisfies Property (FWn) for every n≥0,

3. satisfies Property (FW) if every isometric action of G on a CAT(0) cube complex has a global fixed point.

Property (FW∞) was introduced by Barnhill and Chatterji in [1]^[1], asking the difference between Kazhdan’s Property (T) and Property (FW∞). It turns out that in general Property (T) is quite different from Property (FW) or Property (FW∞). For instance, it is conjectured in [18] that higher rank lattices satisfy Property (FW), but such groups may be far from satisfying Property (T) since some of them are a-T-menable. For positive results in this direction, see [14, Corollary 1.7], [17, Example 6.A.7], [18, Theorem 6.14]. For some (very) recent developments related to Property (FW), see [11, 38, 19].

The second main result of our article shows how to deduce Property (FW∞) from some hyperbolic rigidity. More explicitly:

We emphasise that the property of being hyperbolically elementary is not stable under taking finite-index subgroups, as shown by Example 5.3. Since Thompson’s group V is a simple group, the combination of our two main theorems immediately implies that V satisfies Property (FW∞).

We emphasise that it was previously known that V (as well as F and T) does not act properly on a finite-dimensional CAT(0) cube complex. In fact, since V contains a free abelian group of arbitrarily large rank, it follows that V cannot act properly on any contractible finite-dimensional complex.

Corollary 1.4 contrasts with the known fact that V acts properly on a locally finite infinite-dimensional CAT(0) cube complex. (Indeed, Guba and Sapir showed in [31, Example 16.6] that V coincides with the braided diagram group Db⁢(𝒫,x), where 𝒫 is the semigroup presentation 〈x∣x2=x〉; and Farley constructed in [22] CAT(0) cube complexes on which such groups act.) As a consequence, V provides another negative answer to [1, Question 5.3], i.e., V is a new example of a group satisfying Property (FW∞) but not Property (T). Indeed, as a consequence of [41], a group acting properly on a CAT(0) cube complex does not satisfy Property (T); in fact such a group must be a-T-menable, according to [40], which is a strong negation of Kazhdan’s Property (T).

So V provides an example of a tough transition between finite and infinite dimensions, since on the one hand, V has the best possible cubical geometry in infinite dimension: it acts properly on a locally finite CAT(0) cube complex; and on the other hand, it has the worst possible cubical geometry in finite dimension: every isometric action of V on a finite-dimensional CAT(0) cube complex has a global fixed point. Using the vocabulary of [17], Thompson’s group V satisfies Property PW and Property (FWn) for every n≥0. It seems to be the first such example in the literature.

We would like to emphasise the fact that, although our article is dedicated to Thompson’s group V, we expect that Theorem 1.1 applies to most of the generalisations of V. For instance, without major modifications, our arguments apply to Higman–Thompson groups Vn,r (n≥2, r≥1), to the group of interval exchange transformations IET⁢([0,1]), and to Neretin’s group. However, since there does not exist a common formalism to deal with all the generalisations of V, we decided to illustrate our strategy by considering only V. Therefore, our paper should not be regarded as proving a specific statement about V, but as proposing a general method to prove hyperbolic and cubical rigidities of groups looking like V. In particular, we expect that our strategy works for higher-dimensional Thompson groups.

Finally, we would like to mention that Thompson’s group F is also hyperbolically elementary, since it does not contain any non-abelian free subgroup, but it does not satisfy Property (FW∞) since its abelianisation is infinite. About Thompson’s group T, the situation is less clear, and our strategy does not work. So we leave it as an open question:

The paper is organised as follows. First, Section 2 is dedicated to basic definitions and preliminary lemmas about hyperbolic spaces and CAT(0) cube complexes. In Section 3, we introduce and study a family of particular elements of V, named reducible elements. Finally, in Sections 4 and 5 respectively, we prove our general criteria, namely Theorems 1.1 and 1.3, and we prove Theorem 1.2 and Corollary 1.4 by applying them to V.

2 Preliminaries

2.1 Hyperbolic spaces

In this section, we recall some basic definitions about Gromov-hyperbolic spaces, we fix the notation that will be used in the paper, and we prove a few preliminary lemmas which will be useful later on. For more general information about hyperbolic spaces, we refer to [30, 29, 16, 4].

Definition 2.1.

Let X be a metric space. For every x,y,z∈X, the Gromov product(x,y)z is defined as

12⁢(d⁢(z,x)+d⁢(z,y)-d⁢(x,y)).

Fixing some δ≥0, the space X is δ-hyperbolic if the inequality

(x,z)w≥min⁡((x,y)w,(y,z)w)-δ

is satisfied for every x,y,z,w∈X.

The following definitions will also be needed:

1. A map f:X→Y between two metric spaces is an (A,B)-quasi-isometric embedding, where A>0 and B≥0, if

   1A⋅d⁢(x,y)-B≤d⁢(f⁢(x),f⁢(y))≤A⋅d⁢(x,y)+B

   for every x,y∈X. If f is moreover every point of Y is at distance at most B from the image of f, then f is an (A,B)-quasi-isometry. A quasi-isometry (resp. a quasi-isometric embedding) is a map which an (A,B)-quasi-isometry (resp. an (A,B)-quasi-isometric embedding) for some A>0 and B≥0.

2. Given a metric space X and two constants A>0 and B≥0, an (A,B)-quasigeodesic is an (A,B)-quasi-isometric embedding from a segment of ℝ or ℤ (depending on whether the metric of X is discrete) to X. A quasigeodesic is an (A,B)-quasigeodesic for some A>0 and B≥0.

3. Given a geodesic metric space X and a constant K≥0, a subspace Y⊂X is K-quasiconvex if every geodesic between two points of Y stays in the K-neighbourhood of Y.

4. Given a metric space X and a subspace Y⊂X, the nearest-point projection of a point x∈X onto Y is the set of all the points of Y minimising the distance to x. The nearest-point projection of another subspace Z⊂X onto Y is the union of all the nearest-point projections of the points of Z onto Y.

Usually, it is easier to work with geodesic metric spaces instead of general metric spaces. The following lemma explains a classical trick which allows us to restrict our study to hyperbolic graphs.

Lemma 2.2.

Let X be metric space. If Y denotes the graph whose vertices are the points of X and whose edges link two points at distance at most one, then the inclusion X⊂Y is a (1,0)-quasi-isometry such that any isometry of X extends uniquely to an isometry of Y. As a consequence, if X is hyperbolic, then so is Y.

From now on, all our (hyperbolic) metric spaces will be graphs.

Fixing a graph X, three vertices x,y,z∈X and a geodesic triangle

Δ=[x,y]∪[y,z]∪[z,x],

there exists a unique tripod T and a unique map f:Δ→T such that:

1. f⁢(x),f⁢(y),f⁢(z) are the endpoints of T,

2. f restricts to an isometry on each [x,y], [y,z], [z,x].

The data (T,f) is the comparison tripod of Δ, and the three (not necessarily distinct) points of Δ sending to the center of T define the intriple of Δ.

The following statement is an alternative definition of hyperbolic spaces (among geodesic metric spaces). We refer to the proof of [29, Proposition 2.21] for more information.

Proposition 2.3.

Let X be a δ-hyperbolic graph. For all vertices x,y,z∈X and every geodesic triangle Δ=Δ⁢(x,y,z), if (T,f) denotes the comparison tripod of T, then d⁢(a,b)≤4⁢δ for every a,b∈Δ satisfying f⁢(a)=f⁢(b).

The next statement is a fundamental property satisfied by hyperbolic spaces, often referred to as the Morse Property. See for instance [4, Theorem III.H.1.7].

Theorem 2.4.

Let X be a δ-hyperbolic graph. For every A>0 and every B≥0, there exists some M⁢(δ,A,B), called the Morse constant, such that: for every x,y∈X, any two (A,B)-quasigeodesics between x and y stay at Hausdorff distance at most M⁢(δ,A,B).

Now, let us prove two preliminary lemmas which will be useful in the next sections.

Lemma 2.5.

Let X be a δ-hyperbolic graph and let γ1,γ2 be two lines which are K-quasiconvex for some K≥0. For every x1∈γ1 and x2∈γ2, any geodesic [x1,x2] between x1 and x2 intersects the M⁢(δ,1,2⁢(4⁢δ+K))-neighbourhood of the nearest-point projection of x1 onto γ2.

Proof.

Fix two points x1∈γ1 and x2∈γ2, and a geodesic [x1,x2] between them. Let p∈γ2 be a nearest-point projection of x1 onto γ2. Fixing some geodesics [x1,p] and [p,x2], we claim that [x1,p]∪[p,x2] is a (1,2(4δ+K)-quasigeodesic.

The only point to verify is that, given two points a∈[x1,p] and b∈[p,x2], the inequality

d⁢(a,b)≥d⁢(a,p)+d⁢(p,b)-2⁢(4⁢δ+K)

holds. Let us consider a geodesic triangle Δ=Δ⁢(a,b,p), and let {q1,q2,q3} denote its intriple where q1∈[a,b], q2∈[a,p] and q3∈[b,p]. Notice that, since γ2 is K-quasiconvex, there exists some q∈γ2 satisfying d⁢(q3,q)≤K. One has

d⁢(a,b)=d⁢(a,q1)+d⁢(q1,b)=d⁢(a,q2)+d⁢(b,q3)

=d⁢(a,p)-d⁢(p,q2)+d⁢(p,b)-d⁢(q3,p)

=d⁢(a,p)+d⁢(p,b)-2⁢d⁢(p,q2).

On the other hand,

d⁢(x1,q2)+d⁢(q2,p)=d⁢(x1,γ2)≤d⁢(x1,q)

≤d⁢(x1,q2)+d⁢(q2,q3)+d⁢(q3,q)

≤d⁢(x1,q2)+4⁢δ+K,

hence d⁢(p,q2)≤4⁢δ+K. Our claim follows. We register our conclusion for future use.

Fact 2.6.

Let X be a δ-hyperbolic graph, γ a K-quasiconvex line, and a∈X, b∈γ two vertices. If p∈γ denotes a nearest-point projection of a onto γ, then any concatenation [a,p]∪[p,b] defines a (1,2⁢(4⁢δ+K))-quasigeodesic.

Now, we conclude from the Morse property that the Hausdorff distance between [x1,x2] and [x1,p]∪[p,x2] is at most M⁢(δ,1,2⁢(4⁢δ+K)). The desired conclusion follows. ∎

Lemma 2.7.

Let X be a δ-hyperbolic graph, x,y∈X two vertices and let γ be a K-quasiconvex line. Fix two nearest-point projections x′,y′∈γ respectively of x,y onto γ, and suppose that d⁢(x′,y′)>36⁢δ+5⁢K. Then

d⁢(x,x′)+d⁢(x′,y′)+d⁢(y,y′)-4⁢(6⁢δ+K)≤d⁢(x,y)

≤d⁢(x,x′)+d⁢(x′,y′)+d⁢(y,y′).

Proof.

The right-hand side of our inequality is a consequence of the triangle inequality, so we only have to prove its left-hand side.

Fix some geodesics [x,y], [x′,y′], [x,x′], [y,y′] and [x′,y]. Let {p1,p2,p3} be the intriple of the geodesic triangle Δ⁢(x,y,x′), where p1∈[x,x′], p2∈[x,y], p3∈[x′,y]; and similarly let {q1,q2,q3} be the intriple of the geodesic triangle Δ⁢(x′,y′,y) where q1∈[y,y′], q2∈[x′,y′], q3∈[x′,y]. Notice that, since γ is K-quasiconvex, there exists some q∈γ satisfying d⁢(q2,q)≤K. The configuration is summarised by Figure 1. Notice that

d⁢(y,q1)+d⁢(q1,y′)=d⁢(y,γ)≤d⁢(y,q)

≤d⁢(y,q1)+d⁢(q1,q2)+d⁢(q2,q)

≤d⁢(y,q1)+4⁢δ+K,

hence d⁢(y′,q1)≤4⁢δ+K. Now, we distinguish two cases.

Figure 1

The configuration of points in the proof of Lemma 2.7.

Case 1: Suppose that d⁢(x′,p3)≤d⁢(x′,q3). In this case, there exists some p3′∈[x′,y′] satisfying d⁢(p3,p3′)≤4⁢δ, and because γ is K-quasiconvex, there exists some p3′′∈γ satisfying d⁢(p3′,p3′′)≤K. One has

d⁢(x,p1)+d⁢(p1,x′)=d⁢(x,x′)=d⁢(x,γ)≤d⁢(x,p3′′)

≤d⁢(x,p1)+d⁢(p1,p3)+d⁢(p3,p3′)+d⁢(p3′,p3′′)

≤d⁢(x,p1)+4⁢δ+4⁢δ+K,

hence d⁢(x′,p1)≤8⁢δ+K. Next, notice that

d⁢(x,y)=d⁢(x,p2)+d⁢(p2,y)=d⁢(x,p1)+d⁢(y,p3)

=d⁢(x,x′)-d⁢(x′,p1)+d⁢(y,x′)-d⁢(x′,p3)

=d⁢(x,x′)+d⁢(y,x′)-2⁢d⁢(x′,p1)

and that

d⁢(y,x′)=d⁢(y,q3)+d⁢(q3,x′)=d⁢(y,q1)+d⁢(x′,q2)

=d⁢(y,y′)-d⁢(y′,q1)+d⁢(x′,y′)-d⁢(y′,q2)

=d⁢(y,y′)+d⁢(x′,y′)-2⁢d⁢(y′,q1).

We conclude that

d⁢(x,y)=d⁢(x,x′)+d⁢(x′,y′)+d⁢(y,y′)-2⁢(d⁢(x′,p1)+d⁢(y′,q1))

≥d⁢(x,x′)+d⁢(x′,y′)+d⁢(y,y′)-4⁢(6⁢δ+K).

Case 2: Suppose that d⁢(x′,p3)>d⁢(x′,q3). As a consequence, there exists some q3′∈[x,x′] satisfying d⁢(q3,q3′)≤4⁢δ. Notice that

d⁢(x,q3′)+d⁢(q3′,x′)=d⁢(x,γ)≤d⁢(x,q)

≤d⁢(x,q3′)+d⁢(q3′,q3)+d⁢(q3,q2)+d⁢(q2,q)

≤d⁢(x,q3′)+4⁢δ+4⁢δ+K,

hence d⁢(q3′,x′)≤4⁢(6⁢δ+K). Therefore,

d⁢(x′,y′)≤d⁢(x′,q3′)+d⁢(q3′,q3)+d⁢(q3,q1)+d⁢(q1,y)

≤4⁢(6⁢δ+K)+4⁢δ+4⁢δ+(4⁢δ+K),

i.e., d⁢(x′,y′)≤36⁢δ+5⁢K. This contradicts our assumptions, so our second case cannot happen. ∎

Corollary 2.8.

Let X be a δ-hyperbolic graph, x,y∈X two vertices, [x,y] a geodesic between x and y, and γ a K-quasiconvex line. Fix two nearest-point projections x′,y′∈γ respectively of x,y onto γ, and suppose that d⁢(x′,y′)>36⁢δ+5⁢K. Then d⁢(x,y)≥d⁢(x,x′), so that there exists a unique point a∈[x,y] satisfying d⁢(x,a)=d⁢(x,x′), and moreover d⁢(a,x′)≤2⁢M⁢(δ,1,4⁢(6⁢δ+K)).

Proof.

First of all, notice that, as a consequence of Lemma 2.7, one has

d⁢(x,y)≥d⁢(x,x′)+d⁢(x′,y′)+d⁢(y′,y)-4⁢(6⁢δ+K)

>d⁢(x,x′)+36⁢δ+5⁢K-4⁢(6⁢δ+K)≥d⁢(x,x′),

which proves the first assertion of our statement.

Fix some geodesics [x,x′], [y,y′], [x′,y′]. As a consequence of Fact 2.6 and Lemma 2.7, we know that [x,x′]∪[x′,y′]∪[y′,y] defines a (1,4⁢(6⁢δ+K))-quasigeodesic between x and y. It follows from the Morse property that there exists some a′∈[x,y] satisfying d⁢(x′,a′)≤M⁢(δ,1,4⁢(6⁢δ+K)). One has

d⁢(a,x′)≤d⁢(a,a′)+d⁢(a′,x′)≤d⁢(a,a′)+M⁢(δ,1,4⁢(6⁢δ+K)).

On the other hand,

|d⁢(x,a′)-d⁢(x,x′)|≤d⁢(a′,x′)≤M⁢(δ,1,4⁢(6⁢δ+K)),

so

d⁢(a,a′)=|d⁢(x,a)-d⁢(x,a′)|=|d⁢(x,x′)-d⁢(x,a′)|

≤M⁢(δ,1,4⁢(6⁢δ+K)).

The desired conclusion follows. ∎

Corollary 2.9.

Let X be a δ-hyperbolic graph, γ a K-quasiconvex line and let x,y∈X be two points. If x′,y′∈γ are nearest-point projections onto γ of x,y respectively, then

d⁢(x′,y′)≤d⁢(x,y)+36⁢δ+5⁢K.

Proof.

If d⁢(x′,y′)≤36⁢δ+5⁢K, there is nothing to prove, so we suppose that d⁢(x′,y′)>36⁢δ+5⁢K. As a consequence of Lemma 2.7,

d⁢(x,y)≥d⁢(x′,y′)-4⁢(6⁢δ+K)≥d⁢(x′,y′)-36⁢δ-5⁢K,

which concludes the proof of our corollary. ∎

Now, let X be a δ-hyperbolic graph and g∈Isom⁢(X) an isometry. The translation length of g is

[g]=inf⁡{d⁢(x,g⋅x)∣x∈X};

and the minimal set of g is

Cg={x∈X∣d⁢(x,g⋅x)=[g]}.

It is worth noticing that, because X is a graph, the infimum in the definition of [g] turns out to be a minimum, so that Cg is non-empty.

Definition 2.10.

Let X be a hyperbolic graph and g∈Isom⁢(X) a loxodromic isometry. An axis of g is a concatenation ℓ=⋃k∈ℤgk⋅[x,g⋅x] for some x∈Cg.

Noticing that an axis of g is a [g]-local geodesic, the following lemma follows from [4, Theorem III.Γ.1.13].

Lemma 2.11.

Let X be a δ-hyperbolic graph and let g∈Isom⁢(X) be a loxodromic isometry satisfying [g]>32⁢δ. Any axis of g is 12⁢δ-quasiconvex.

We conclude this section with a last preliminary lemma, which will be fundamental in the proof of the hyperbolic rigidity of Thompson’s group V.

Lemma 2.12.

Let X be a hyperbolic graph and g,h∈Isom⁢(X) two isometries. Suppose that g is loxodromic of translation length at least 525⁢δ and that h is elliptic. Fix an axis ℓ of g. If hg is elliptic, then there exists a point x∈ℓ such that

d⁢(x,h⁢x)≤8⁢M⁢(δ,1,62⁢δ)+243⁢δ.

Proof.

For convenience, fix a G-equivariant map π:X→ℓ sending every point of X to one of its nearest-point projections, and set M=2⁢M⁢(δ,1,62⁢δ). Because h is elliptic, we know from [4, Lemma III.Γ.3.3] that there exists some x∈X such that 〈h〉⋅x has diameter at most 17⁢δ.

Suppose that there exists some y∈X such that the distances d⁢(π⁢(x),π⁢(y)) and d⁢(π⁢(h⁢y),π⁢(h⁢x)) are both greater than 96⁢δ. Fix a geodesic [x,y]. We know from Corollary 2.8 that there exists a point z∈[x,y] satisfying d⁢(x,z)=d⁢(x,π⁢(x)) such that d⁢(z,π⁢(x))≤2⁢M. Similarly, we know from Corollary 2.8 that there exists a point w∈h⋅[x,y] satisfying d⁢(h⁢x,w)=d⁢(h⁢x,π⁢(h⁢x)) such that

d⁢(w,π⁢(h⁢x))≤2⁢M.

Notice that

d⁢(h⁢z,w)=|d⁢(h⁢x,w)-d⁢(h⁢x,h⁢z)|=|d⁢(h⁢x,π⁢(h⁢x))-d⁢(x,π⁢(x))|

≤d⁢(x,h⁢x)+d⁢(π⁢(x),π⁢(h⁢x))≤130⁢δ,

where the last inequality is justified by Corollary 2.9. Consequently,

d⁢(z,h⁢z)≤d⁢(z,π⁢(x))+d⁢(π⁢(x),π⁢(h⁢x))+d⁢(π⁢(h⁢x),w)+d⁢(w,h⁢z)

≤2⁢M+(17⁢δ+96⁢δ)+2⁢M+130⁢δ=4⁢M+243⁢δ.

We conclude that π⁢(x) is a point satisfying the conclusion of our lemma, since

d⁢(π⁢(x),h⁢π⁢(x))≤d⁢(z,h⁢z)+2⁢d⁢(z,π⁢(x))≤8⁢M+243⁢δ.

Next, suppose that for every y∈X satisfying d⁢(π⁢(x),π⁢(y))>96⁢δ one has

d⁢(π⁢(h⁢y),π⁢(h⁢x))≤96⁢δ.

Consequently, since

d⁢(π⁢(x),π⁢(g⁢x))=d⁢(π⁢(x),g⁢π⁢(x))=[g]>96⁢δ,

it follows that d⁢(π⁢(h⁢x),π⁢(h⁢g⁢x))≤96⁢δ. So

d⁢(π⁢(g⁢x),π⁢(g⁢h⁢g⁢x))=d⁢(π⁢(x),π⁢(h⁢g⁢x))

≤d⁢(π⁢(x),π⁢(h⁢x))+d⁢(π⁢(h⁢x),π⁢(h⁢g⁢x))

≤d⁢(x,h⁢x)+96⁢δ+96⁢δ≤209⁢δ,

where the first inequality of the second line is justified by Corollary 2.9. Next, since

d⁢(π⁢(x),π⁢(g⁢h⁢g⁢x))≥d⁢(π⁢(x),π⁢(g⁢x))-d⁢(π⁢(g⁢x),π⁢(g⁢h⁢g⁢x))

≥[g]-209⁢δ>96⁢δ,

we deduce from Lemma 2.7 that

d⁢(x,g⁢h⁢g⁢x)≥d⁢(x,π⁢(x))+d⁢(π⁢(x),π⁢(g⁢h⁢g⁢x))+d⁢(π⁢(g⁢h⁢g⁢x),g⁢h⁢g⁢x)-72⁢δ.

By noticing that

d(π(x),π(ghgx))≥d(π(x),π(gx))-d(π(gx),π(ghgx)≥[g]-209δ

and that

d⁢(π⁢(g⁢h⁢g⁢x),g⁢h⁢g⁢x)=d⁢(π⁢(h⁢g⁢x),h⁢g⁢x)≥d⁢(h⁢g⁢x,x)-d⁢(x,π⁢(h⁢g⁢x))

≥d⁢(x,h⁢g⁢x)-d⁢(x,π⁢(x))-d⁢(π⁢(x),π⁢(h⁢g⁢x))

≥d⁢(x,h⁢g⁢x)-d⁢(x,π⁢(x))-209⁢δ,

the previous inequality becomes

d⁢(x,g⁢h⁢g⁢x)≥d⁢(x,π⁢(x))+[g]-209⁢δ+d⁢(x,h⁢g⁢x)

-d⁢(x,π⁢(x))-209⁢δ-72⁢δ

≥d⁢(x,h⁢g⁢x)+[g]-490⁢δ,

hence

d⁢(x,(h⁢g)2⁢x)≥d⁢(x,g⁢h⁢g⁢x)-d⁢(x,h⁢x)≥d⁢(x,h⁢g⁢x)+[g]-507⁢δ.

Since [g]>525⁢δ by assumption, it follows that

d⁢(x,(h⁢g)2⁢x)>d⁢(x,h⁢g⁢x)+18⁢δ.

According to [29, Corollaire 8.22], this inequality implies that hg is loxodromic, contradicting our hypotheses. ∎