Well displacing representations and orbit maps

Thomas DELZANT^{∗}  Olivier GUICHARD^{†}
François LABOURIE ^{‡}  Shahar MOZES^{§}

March 30, 2007

1 Introduction

1.1 Displacement function

Let γ be an isometry of a metric space X. We recall that the displacement of γ is

d_{X}(γ) = inf_{x∈X} d(x, γ(x)).

We observe that d_{X}(γ) is an invariant of the conjugacy class of γ.

If C is the Cayley graph of a group Γ with set of generators S and word length ∥ ∥_{S}, the displacement function is called the translation length and is denoted by ℓ

ℓ_{S}(γ) = inf_{η} ∥ηγη^{−1}∥_{S}.

We finally say the action by isometries on X of a group Γ is well displacing, if given a set S of generators of Γ, there exist positive constants A and B such that

d_{X}(γ) ⩾ Aℓ_{S}(γ) − B.

This definition does not depends on the choice of S. As first examples, It is easy to check that cocompact groups are well displacing, as well as convex cocompact whenever X is Hadamard (i.e. complete, non positively curved and simply connected). We recall that a cocompact action is by definition a properly discontinuous action whose quotient is compact, and a convex cocompact is an action such that there exists a convex invariant on which the action is cocompact.

∗IRMA et Université L. Pasteur, 7 rue Descartes, F-67084 Strasbourg

†CNRS, Orsay cedex, F-91405; Univ. Paris-Sud, Laboratoire de Mathématiques, Orsay F-91405 Cedex;

‡Univ. Paris-Sud, Laboratoire de Mathématiques, Orsay F-91405 Cedex; CNRS, Orsay cedex, F-91405

§Institute of Mathematics, Hebrew University, Jerusalem, Israel

The notion naturally arose in [7] where it is shown that for well displacing representations, the energy functional is proper on Teichmüller space and that moreover a large class of representations of surface groups are well displacing.

The purpose of this note is to investigate the relation of well displacing actions, to actions for which are the orbit maps are quasi isometric embeddings. For hyperbolic groups the two notions turn out to be equivalent. However in general, the answer is a little more involved.

• There exist infinite groups whose actions are always well displacing. However, an infinite group do have actions for which orbit maps are not quasi isometric embeddings: the trivial action. Therefore, in general, well displacing representation do not have orbit maps which are quasi isometric embeddings.

• There exist a class of group for which if an orbit map is a quasi isometric embedding then the action is well displacing. The main example is the class of hyperbolic groups. In general these are the groups, for which the stable length is equal to to the translation length (cf. Proposition 3.0.2).

• There exists a class of group for which every well displacing action have orbit maps which are quasi isometric embeddings. This class of group are those which are undistorted in their conjugacy classes (see Section 4) – or U-property – reminiscent of a statement of Abels, Margulis and Soifer about proximal maps [1]. We prove in Theorem 4.1.1, that some class of linear groups – in particular lattices — enjoys the U-property. We also prove that hyperbolic groups have the U-property.

• There exists action of groups – which have the U-property – whose orbit maps are quasi isometric embeddings but which are not well displacing. The simplest example is SL(n, ℤ) acting on SL(n, ℝ)/SO(n, ℝ). More generally in Corollary 5.2.3, we show that if a group have infinite contortion (see Section 5.1) – in particular every residually finite group – then any linear representation which contains a unipotent is not well displacing.

Finally, we investigate in a complement a property related to infinite contortion which we call the bounded depth roots property (see Section 5.4). We finish by a list of questions

1. Do all linear groups have U-property?

Contents

1 Introduction  1
1.1 Displacement function . . . . . . . . . . . . . . . . . . . . . . . . 1
2 Infinite groups whose actions are always well displacing  3
3 Orbit maps which are quasi isometry implies well displacing  3

4 Groups whose well displacing actions have orbit maps which are quasi isometric embeddings 4
4.1 Linear Groups having U-property . . . . . . . . . . . . . . . . . . 5
4.1.1 Generalities for U-property . . . . . . . . . . . . . . . . . 6
4.1.2 Proximality . . . . . . . . . . . . . . . . . . . . . . . . . . 7
4.1.3 Proximal elements and translation lengths . . . . . . . . . 8
4.1.4 Proof of Theorem 4.1.1 . . . . . . . . . . . . . . . . . . . 9
4.2 Hyperbolicity and the U-property . . . . . . . . . . . . . . . . . . 9
4.2.1 Almost cyclically reduced elements . . . . . . . . . . . . . 10
4.2.2 Ping pong pairs . . . . . . . . . . . . . . . . . . . . . . . . 11
4.2.3 Proof of Proposition 4.2.2 . . . . . . . . . . . . . . . . . . 12
5 Non well displacing actions whose orbit maps are quasi isometric embeddings 14
5.1 Infinite contortion . . . . . . . . . . . . . . . . . . . . . . . . . . 14
5.2 Displacement function and infinite contortion . . . . . . . . . . . 14
5.3 Non uniform lattices . . . . . . . . . . . . . . . . . . . . . . . . . 16
5.4 Bounded depth roots . . . . . . . . . . . . . . . . . . . . . . . . . 16
5.4.1 Bounded depth roots property for SL(n, ℤ) . . . . . . . . 17
5.4.2 Commensurability . . . . . . . . . . . . . . . . . . . . . . 18
5.4.3 Proof of Lemma 5.4.2. . . . . . . . . . . . . . . . . . . . . 18

2 Infinite groups whose actions are always well displacing

We have

Proposition 2.0.1 There exists infinite finitely generated groups whose actions are always well displacing. Hence there exists action for which the orbit maps are not quasi isometric embeddings, but which are well displacing.

Proof : Denis Osin [9] as constructed infinite finitely generated groups with exactly n conjugacy classes. Any action of such a group is well displacing. For the second part, we just take the trivial action on a point.

3 Orbit maps which are quasi isometry implies well displacing

If a group Γ acts by isometries on a metric space X, we define the stable norm with respect to X by

[g]_{∞}^{X} = lim inf_{n→∞} \frac{1}{n}d(x_{0}, g^{n}(x_{0})).

We observe that this quantity does not depends on the choice of the base point x. The stable norm [g]_{∞} is the stable norm with respect to the Cayley graph of Γ.

We now prove the following easy result Proposition 3.0.2 If a group Γ is such that the exist α > 0 such that

∀g ∈ Γ, [g]_{∞} ⩾ α.ℓ(g),

then every action of Γ on (X, d) for which the orbit map is a quasi isometric embedding is well displacing.

Proof : by definition, if the orbit map is a quasi isometric embedding, for every x ∈ X there exists some constant A and B such that

A∥γ∥ + B ⩾ d(x, γ(x)) ⩾ A^{−1}∥γ∥ − B.

It follows that

A[γ]_{∞} ⩾ [γ]_{∞}^{X} ⩾ A^{−1}[γ]_{∞}.

Now we remark that

[γ]_{∞}^{X} ⩽ d_{X}(γ).

The result follows.Q.e.d.

Remarks:

• We wil show later on that this inequality fails for SL(n, ℤ) for n ⩾ 3.

• On the other hand, we observe following [4, p 119], that for Γ a hyperbolic group, the stable norm of an element coincides up to a constant with its translation length.

Therefore, we have

Corollary 3.0.3 Let Γ be a hyperbolic group. If an isometric action is such that the orbit maps are quasiisometries, then this action is well displacing.

4 Groups whose well displacing actions have orbit maps which are quasi isometric embeddings

We say a finitely generated group is undistorted in its conjugacy classes – in short, has U-property – , if there exists finitely many elements g_{1}, … g_{p} of Γ, positive constants A and B such that

∀γ ∈ Γ, ∥γ∥ ⩽ A sup_{1⩽i⩽n} ℓ(g_{i}γ) + B.

Remarks:

• This property is clearly independent of S.

• This property is satisfied by free groups and commutative groups. On the other hands the groups constructed by D. Osin [9] described in the above paragraph do not have this U-property.

• Note also that this property is very similar to the statement of Abels, Margulis and Soifer’s result [1, Theorem 4.1] and it is no surprise that their result plays a role in the proof of Theorem 4.1.1.

• Finally, by the conjugacy invariance of the translation length, ℓ(g_{i}γ) = ℓ(γg_{i}) so we will indifferently write this property with left or right multiplication by the finite family (g_{i}).

Lemma 4.0.4 If Γ has U-property, then every well displacing action have orbit maps which are quasi isometric embeddings.

Proof : Indeed, assume that Γ acts on X by isometries and that the action is well displacing. In particular we have positive constants α and β so that

∀x ∈ X, γ ∈ Γ, d(x, γ(x)) ⩾ αℓ_{S}(γ) − β.

Moreover there exists finitely many elements g_{1}, … g_{p} of Γ, positive constants A and B such that

∀γ ∈ Γ, A∥γ∥ − B ⩽ sup_{1⩽i⩽n} ℓ(g_{i}γ).

Hence, let x ∈ X, then

d(x, γ(x)) ⩾ sup_{1⩽i⩽n} d(x, g_{i}.γ(x)) − sup_{1⩽i⩽n} d(x, g_{i}(x))
⩾ α sup_{1⩽i⩽n} ℓ(g_{i}γ) − β − sup_{1⩽i⩽n} d(x, g_{i}(x))
⩾ α.A∥γ∥ − Bα − β − sup_{1⩽i⩽n} d(x, g_{i}(x)).

Hence the orbit map is a quasi isometric embedding. Q.e.d.

We will prove in the next section,

Theorem 4.0.5 Every uniform lattice – and non uniform lattice in higher rank – in characteristic zero has U-property. In particular every surface group has U-property.

Moreover, we show that hyperbolic groups have U-property.

4.1 Linear Groups having U-property

We prove the following result which implies Theorem 4.0.5

Theorem 4.1.1 Let Γ be a finitely generated group and G a reductive group defined over a field F, suppose that

• there exists a homomorphism Γ → G(F) with Zariski dense image,

• there are a finitely many field homomorphisms (i_{v})_{v∈S} of F in local fields F_{v} such that the diagonal embedding Γ → Π_{v∈S}G(F_{v}) a quasi-isometric embedding.

Then the group Γ has U-property.

Since lattices are Zariski dense (Borel theorem) and that higher rank irreducible lattices in characteristic zero are quasi isometrically embedded ([8]), this theorem implies Theorem 4.0.5 for higher rank lattices. The same holds for all uniform lattices. A specific corollary is the following

Corollary 4.1.2 Let Γ be a finitely generated that is quasi isometrically embedded and Zariski dense in a reductive group G(F) where F is a local field. Then Γ has U-property.

4.1.1 Generalities for U-property

We prove two lemmas for the U-property.

Lemma 4.1.3 . Let Γ be a finitely generated group. Let Γ_{0} ◁ Γ be a normal subgroup of finite index. If Γ_{0} has U-property, so has Γ.

We do not know whether the converse statement holds, in other words whether U-property is a property of commensurability classes.

Proof : We first observe that every finite index subgroup of a finitely generated group is finitely generated. Let S_{0} be a generating set for Γ_{0} and write Γ as the union of left cosets for Γ_{0}

Γ = ⋃_{t∈T} Γ_{0} · t.

We assume that T is symmetric. Clearly S = S_{0} ∪ T is a generating set for Γ. We denote ∥ · ∥_{Γ_{0}} and ℓ_{Γ_{0}} the word and translation lengths for Γ_{0}. We observe that Γ_{0} is quasi-isometrically embedded in Γ. Hence there exist

positive constants α and β such that

∀γ ∈ Γ_{0}, ∥γ∥_{Γ_{0}} ⩾ ∥γ∥_{Γ} ⩾ α∥γ∥_{Γ_{0}} − β.  (1)

For any γ in Γ, we write γ = γ_{0}t_{0} with t_{0} ∈ T and γ_{0} ∈ Γ_{0}. Hence

∥γ∥_{Γ} ⩽ ∥γ_{0}∥_{Γ_{0}} + 1 ⩽ A sup ℓ_{Γ_{0}}(γ_{0}g_{i}) + B + 1

since Γ_{0} has U-property.

Finally we need to compare ℓ_{Γ} and ℓ_{Γ_{0}}. Let δ in Γ_{0}, then

ℓ_{Γ}(δ) = inf_{t∈T,η∈Γ_{0}} ∥tηtδη^{−1}t^{−1}∥_{Γ}
⩾ inf_{η∈Γ_{0}} ∥ηδη^{−1}∥_{Γ} − 2
⩾ α inf_{η∈Γ_{0}} ∥ηδη^{−1}∥_{Γ_{0}} − β − 2
⩾ αℓ_{Γ_{0}}(δ) − β − 2  (2)

Finally combining Inequalities (4.1.1) and (2) , we have

∥γ∥_{Γ} ⩽ \frac{A}{α} sup_{t∈T,i∈I} ℓ_{Γ}(γt^{−1}β_{i}) + B + 1 + \frac{β + 2}{α}.

This is exactly the U-property for Γ. Q.e.d.

Also

Lemma 4.1.4 Suppose that Γ → Γ_{0} is onto with finite kernel and Γ finitely generated. Then the group Γ has U-property if and only if Γ_{0} has.

Proof : We choose a generating set S for Γ which contains the kernel of Γ → Γ_{0}. We choose the generating set S_{0} for Γ_{0} to be the image of S. Then we have, using surjectivity, for all γ projecting to γ_{0}

∥γ∥_{Γ} ⩾ ∥γ_{0}∥_{Γ_{0}} ⩾ ∥γ∥_{Γ} − 1,
ℓ_{Γ}(γ) ⩾ ℓ_{Γ_{0}}(γ_{0}) ⩾ ℓ_{Γ}(γ) − 1.

These two inequalities enable us to transfer the U-property from Γ to Γ_{0} and vice versa. Q.e.d.

4.1.2 Proximality

We recall the notion of proximality and a result of Abels, Margulis and Soifer.

Let k be a local field. Let V be a finite dimensional k-vector space equipped with a norm. Let d be the induced metric on ℙ(V). Let r and ϵ be positive numbers such that

r > 2ϵ.

An element g of SL(V) is said to be (r, ε)-proximal, if there exist a point x_{+} in ℙ(V) and an hyperplane H in V such that

• d(x_{+}, ℙ(H)) ⩾ r,

• ∀x ∈ ℙ(V), d(x, ℙ(H)) ⩾ ε ⟹ d(g · x, x_{+}) ⩽ ε.

In particular a proximal element has a unique eigenvalue of highest norm and conversely for an element g admitting a unique eigenvalue of highest norm some power of g is proximal (for some (r, ϵ)).

We cite the needed result from [1] and [2].

Theorem 4.1.5 ([1] Theorem 5.17) Let G a semisimple group over a field F. Let (i_{v})_{v∈V} be finitely many field homomorphisms of F in local fields F_{v}. Let ρ_{v} : G(F_{v}) → GL(n_{v}, F_{v}) be an irreducible representation of G(F_{v}) for each v.

Suppose that Γ is a Zariski dense subgroup of G(F). Suppose that for every v, ρ_{v}(Γ) contains proximal elements. Then there exist

• r > 2ε > 0

• a finite subset Δ ⊂ Γ,

such that for every γ in Γ there is some δ in Δ such that ρ_{v}(γs) is (r, ε)-proximal for every v in V.

Note that the usual statement is when working with one local field but the proof of the extension of this result and the following to our settings is straightforward.

We shall also need the following Lemma

Lemma 4.1.6 ([2] Corollaire p.13) Let Γ be a Zariski dense subgroup in G(k), G a reductive group over a local field k. Then Γ is unbounded if and only if there exists an irreducible representation of G(k), such that ρ(Γ) contains a proximal element.

The special case of k = ℝ was proved in [3].

4.1.3 Proximal elements and translation lengths

We recall some facts on length and translation length in G(k) where G is a semisimple group over a local field k.

Let K be the maximal compact subgroup of G(k). This defines a norm ∥g∥_{G} = d_{G/K}(K, gK) in G(k) which satisfies

∥gh∥_{G} ⩽ ∥g∥_{G} + ∥h∥_{G}.

We also consider the translation length ℓ_{G} in G(k):

ℓ_{G}(g) = inf_{h∈G} ∥hgh^{−1}∥_{G}.

Observe that the translation norm is actually independent of the choice of the maximal compact subgroup since they are all conjugated. The translation length and norm of (r, ϵ)-proximal elements can be compared :

Lemma 4.1.7 (compare [2]§ 4.5) Let G be a semisimple group over a local field k. Let ρ : G(k) → GL(n, k) be an irreducible representation. Let ε > 0.

Then there exist positive constants α and β such that if ρ(g) is (r, ε)-proximal then

ℓ_{G}(g) ⩾ α∥g∥_{G} − β  (3)

Proof : We use classical notation and refer to [2, p.6–8] for precise definitions.

Any g in G(k) is contains in a unique double coset Kμ(g)K. The element μ(g) ∈ A^{+} ⊂ A is called the Cartan projection of g. Here we see A^{+} as a subset of a cone A^{×} in some R-vector space.

For some integer n (n = 1 in the archimedean case) the element g^{n} admits a Jordan decomposition g^{n} = g_{e}g_{h}g_{u} with g_{e}, g_{h}, g_{u} commuting, g_{e} elliptic, g_{u} unipotent and g_{h} hyperbolic, i.e. conjugated to a unique element a ∈ A^{+}. We set λ(g) = \frac{1}{n}a ∈ A^{×}.

If we fix some norm on the vector space containing the cone A^{×} then (up to quasi isometry constants) the norm of μ(g) is ∥g∥_{G} in G(k) and the norm of λ(g) is ℓ_{G}(g).

Then by a result of Y. Benoist [2], there exists a compact subset N_{ε} of the vector space containing A^{×} such that for every g such that ρ(g) is (r, ε)-proximal we have

λ(g) − μ(g) ∈ N_{ε}.

The lemma follows. Q.e.d.

4.1.4 Proof of Theorem 4.1.1

By taking a finite index normal subgroup and projecting (Lemmas 4.1.3 and 4.1.4) we can make the hypothesis that G is the product of a semisimple group S and a torus T and Γ is a subgroup of S(F) × T(F).

Moreover, since length and translation length for elements in T(F_{v}) are equal, we only need to work with the semisimple part S.

Finally, it suffices to prove the existence of a finite family F ⊂ Γ and constants A, B such that, for any γ ∈ Γ

∥γ∥_{S} ⩽ A sup_{f∈F} ℓ_{S}(γf) + B,  (4)

where S = Π_{v∈V} S(F_{v}). Indeed, since Γ is quasiisometrically embedded in G, ∥γ∥_{Γ} is less than ∥γ∥_{G} = ∥γ∥_{T} +∥γ∥_{S} and ℓ_{G}(γ) = ℓ_{T} (γ)+ℓ_{S}(γ) = ∥γ∥_{T} +ℓ_{S}(γ) is less than ℓ_{Γ}(γ) –up to quasi isometry constants– and Inequality 4 implies that Γ has U-property.

Note that we can forget any completion F_{v} where the subgroup Γ ⊂ S(F_{v}) is bounded without changing the fact that Γ is quasiisometrically embedded. So by Lemma 4.1.6, for each v there is an irreducible representation ρ_{v} : S(F_{v}) → GL(n_{v}, F_{v}) such that ρ_{v}(Γ) contains proximal elements.

Applying Theorem 4.1.5 we find a finite family F ⊂ Γ and r, ε such that for every γ in Γ there is some f ∈ F such ρ_{v}(γf) is (r, ε)-proximal for every v. Hence, as consequence of Lemma 4.1.7, we have for such γ and f :

∥γf∥_{S} ⩽ Aℓ_{S}(γf) + B.

This implies Inequality (4) and concludes the proof.

4.2 Hyperbolicity and the U-property

Let Γ be a finitely generated group and S a set of generators. Let d be its word distance and ∥g∥ = d(e, g). We denote by

⟨g, h⟩_{u} = \frac{1}{2}(d(g, u) + d(h, u) − d(g, h)),

the Gromov product – based at the u – on Γ. We abbreviate ⟨g, h⟩_{e} by ⟨g, h⟩. Observe that

⟨gu, hu⟩_{u} = ⟨g, h⟩_{e}.  (5)

Recall that Γ is called δ-hyperbolic if for all g, h, k in Γ we have

⟨g, k⟩ ⩾ inf(⟨g, h⟩; ⟨h, k⟩) − δ  (6)

and Γ is called hyperbolic if it is δ-hyperbolic for some δ. A hyperbolic group is called non elementary if it is not finite and does not contain ℤ as a subgroup of finite index.

Then

Proposition 4.2.1 Hyperbolic groups have U-property.

We recall the stable translation length of an element g:

[g]_{∞} = lim_{n→∞} \frac{∥g^{n}∥}{n}.

We remark that obvioulsy

[g]_{∞} ⩽ ℓ(g).

We shall actually prove

Proposition 4.2.2 Let Γ be hyperbolic. There exist a pair u, v ∈ Γ and a constant α such that for every g one has

∥g∥ ⩽ 3 sup([g]_{∞}, [gu]_{∞}, [gv]_{∞}) + α.

In particular Γ has property U.

Remarks:

• Let Γ be a free group generated by some elements u, v, w_{1}, … w_{n}. Then G is 0 hyperbolic. If [g] ≠ ∥g∥ the first letter of g must be equal to the inverse of the last one. Multipliying either by u or by v we find a new element which is cyclically reduced : for this element the stable translation length and the length are equal. The proof of Proposition 4.2.2 is a generalisation of this remark.

• More generally, the proof shows that if a group acts by isometry on some hyperbolic space with orbit maps which are quasiisometries, if furthermore, the group contains a ping pong pair in the sense of Definition 4.2.2 then the group have the U-property.

4.2.1 Almost cyclically reduced elements

An element g in Γ is said to be almost cyclically reduced if ⟨g, g^{−1}⟩ ⩽ \frac{∥g∥}{3} − δ We prove in this paragraph

Lemma 4.2.3 If g is almost cyclically reduced , then

[g]_{∞} ⩾ \frac{∥g∥}{3}.

The following result [5, Lemme 1.1] will be useful. Lemma 4.2.4 Let (x_{n}) be a finite or infinite sequence in G. Suppose that

d(x_{n+2}, x_{n}) ⩾ sup(d(x_{n+2}, x_{n+1}), d(x_{n+1}, x_{n})) + a + 2δ,

or equivalently that

⟨x_{n+2}, x_{n}⟩_{x_{n+1}} ⩽ \frac{1}{2} inf(d(x_{n+2}, x_{n+1}), d(x_{n+1}, x_{n})) − \frac{a}{2} − δ.

Then

d(x_{n}, x_{p}) ⩾ |n − p|a.

This implies Lemma 4.2.3. Proof : Let x_{n} = g^{n}. By left invariance and since g is almost cyclically reduced,

⟨x_{n+2}, x_{n}⟩_{x_{n+1}} = ⟨g, g^{−1}⟩ ⩽ \frac{∥g∥}{2} − \frac{a}{2} − δ,

for a = \frac{∥g∥}{3}. By Lemma 4.2.4,

∥g^{n}∥ ⩾ n\frac{∥g∥}{3}.

The result follows Q.e.d. 4.2.2 Ping pong pairs A ping pong pair in Γ is a pair of elements u, v such that :

1. inf(∥u∥, ∥v∥) ⩾ 100δ.

2. ⟨u^{±1}, v^{±1}⟩ ⩽ \frac{1}{2} inf(∥u∥, ∥v∥) − 20δ

3. ⟨u, u^{−1}⟩ ⩽ \frac{∥u∥}{2} − 20δ and ⟨v, v^{−1}⟩ ⩽ \frac{∥v∥}{2} − 20δ Remarks:

• A ping pong pair generates a free subgroup. This is an observation from [5]. To prove this, consider a reduced word w on the letter u, v, u^{−1}, v^{−1}. If x_{n} is the prefix of length n of w, the sequence x_{n} satisfies the hypothesis of Lemma 4.2.4.

• In the present proof, the third property will not be used. We shall prove Lemma 4.2.5 If Γ is hyperbolic non-elementary, there exists a ping pong pair.

Proof : In [6] explicit ping pong pairs are constructed. Here is a construction whose idea goes back to F. Klein. Let f be some hyperbolic element (an element of infinite order). Replacing f by a conjugate of some power, we may assume that

∥f∥ = [f] > 1000δ.

As Γ is not elementary, there exists a generator a of Γ which do not fix the pair of fixed points f^{+}, f^{−} of f on the boundary ∂Γ : otherwise, since the action of Γ is topologically transitive, ∂Γ would be reduced to these two points and Γ would be elementary. Now, let us prove that for some integer N, (f, af^{N}a^{−1}) is a ping pong pair. We have

lim_{N→+∞} f^{N} = f^{+} ≠ af^{+} = lim_{N→+∞} af^{N}a^{−1}.

It follows that the Gromov product ⟨f^{N}, af^{N}a^{−1}⟩ remains bounded, by the very definition of the boundary. Hence, for N large enough, we have

(f, af^{N}a^{−1}) ⩽ \frac{1}{2} inf(∥f^{N}∥, ∥af^{N}a^{−1}∥) − 20δ.

A similar argument also yields that ⟨f^{N}, af^{−N}a^{−1}⟩ remains bounded. Therefore (f^{N}, af^{N}a^{−1}) is a ping pong pair for N ≫ 1. Q.e.d.

4.2.3 Proof of Proposition 4.2.2

We first reduce this proof to the following Lemma

Lemma 4.2.6 Let (u, v) be a ping pong pair. Let g ∈ Γ such that

∥g∥ ⩾ 3 sup(∥u∥, ∥v∥) + 100δ.

Then one of the three elements g, gu, gv is almost cyclically reduced.

We oberve at once that Proposition 4.2.2 follows from Lemma 4.2.3, 4.2.5 and 4.2.6 : choose a ping pong pair u, v and take

α = 3 sup(∥u∥, ∥v∥) + 100δ.

Proof : Assume g is not almost cyclically reduced. Then

⟨g, g^{−1}⟩ ⩾ \frac{∥g∥}{3} − δ ⩾ sup(∥u∥, ∥v∥) + 30δ.  (7)

Moreover one of the following pair of inequalities hold

⟨g^{−1}, u^{±1}⟩ ⩽ \frac{∥u∥}{2} − 10δ,  (8)

or

⟨g^{−1}, v^{±1}⟩ ⩽ \frac{∥v∥}{2} − 10δ,  (9)

Otherwise, by the definitionof hyperbolicity we would have for some ε, ε′ ∈ {±1} ,

⟨u^{ε}, v^{ε′}⟩ ⩾ \frac{1}{2} inf(∥u∥, ∥v∥) − 10δ − δ,

hence contradicting the second property of the definition of ping pong pairs.

So we may assume that Inequality 8 holds. We will show that gu is almost cyclically reduced. Let k = gu. By the triangle inequality ⟨u^{−1}, g⟩ ⩽ ∥u∥. Thus from Inequality 7, we deduce that

inf(⟨g, u^{−1}⟩, ⟨g, g^{−1}⟩) = ⟨g, u^{−1}⟩.

Then, by the definition of hyperbolicity, we get:

⟨g, u^{−1}⟩ ⩽ ⟨u^{−1}, g^{−1}⟩ + δ.  (10)

Using Inequality 8 now, we have:

⟨g, u^{−1}⟩ ⩽ \frac{∥u∥}{2} − 9δ.  (11)

Note that

⟨g, k⟩ = \frac{1}{2}(∥g∥ + ∥gu∥ − ∥u∥) ⩾ \frac{1}{2}(2∥g∥ − 2∥u∥) ⩾ 2∥u∥ + 100δ.

By the triangle inequality again

⟨k, u^{−1}⟩ ⩽ ∥u∥.

Therefore

inf(⟨g, k⟩, ⟨k, u^{−1}⟩) = ⟨k, u^{−1}⟩.

From hyperbolicity and Inequality 11, we have

⟨k, u^{−1}⟩ ⩽ ⟨g, u^{−1}⟩ + δ ⩽ \frac{∥u∥}{2} − 8δ  (12)

Applying successively Inequalities 8 and 12, we get that

⟨k^{−1}, u^{−1}⟩ = ∥u∥ − ⟨u, g^{−1}⟩ ⩾ \frac{∥u∥}{2} + 10δ ⩾ ⟨k, u^{−1}⟩ + 18δ.  (13)

By hyperbolicity,

inf(⟨k, k^{−1}⟩, ⟨k^{−1}, u^{−1}⟩) ⩽ ⟨k, u^{−1}⟩ + δ.

Therefore, Inequalities 12 and 13 imply that

⟨k, k^{−1}⟩ ⩽ ⟨k, u^{−1}⟩ + δ ⩽ \frac{∥u∥}{2} − 7δ  (14)

Since ∥k∥ ⩾ ∥g∥ − ∥u∥ ⩾ 3∥u∥ − ∥u∥ + 100δ, we finally obtain that k is almost cyclically reduced. Q.e.d.

5 Non well displacing actions whose orbit maps are quasi isometric embeddings

We prove in particular

Proposition 5.0.7 The action of SL(n, ℤ) on X_{n} = SL(n, ℝ)/SO(n, ℝ) is not well displacing, although, for n ⩾ 3, the orbit maps are quasi isometric embeddings.

The second part of this statement is a theorem of Lubotzky, Mozes and Ragunathan [8]. Note that for the action of SL(2, ℤ) on the hyperbolic plane ℍ_{2} = X_{2} the orbit maps are not quasi isometries so it is obviously not well displacing since SL(2, ℤ) is a hyperbolic group (see Corollary 3.0.3).

5.1 Infinite contortion

We say a group Γ has infinite contortion, if the set of conjugacy classes of powers of every non torsion element is infinite. In other words, for every non torsion element γ, for every finite family g_{1}, …, g_{q} of conjugacy classes of elements of Γ, there exists k > 0 such that

∀i ∈ {1, …, q}, γ^{k} ∉ g_{i}.

We prove

Lemma 5.1.1 Every residually finite group has infinite contortion.

Proof : Let γ be a non torsion element. Let g_{1}, …, g_{n} be finitely many conjugacy classes. We want to prove that there exists k > 0 such that γ^{k} belongs to no g_{i}. Since γ is not a torsion element, we can assume that all the g_{i} are non trivial. Let h_{i} ∈ g_{i}. Since all h_{i} are non trivial, by residual finiteness there exist a homomorphism ϕ in a finite groups H, such that

∀i, φ(h_{i}) ≠ 1.

Let k = ∥H∥, hence φ(γ^{k}) = 1. This implies that γ^{k} ∉ g_{i} Q.e.d.

5.2 Displacement function and infinite contortion

We will prove

Lemma 5.2.1 Assume Γ has infinite contortion. Assume Γ acts cocompactly and properly discontinuoulsy by isometry on a space X. Assume furthermore that every closed bounded set in X is compact. Then, for every non torsion element γ in Γ, we have

lim sup_{p→∞} d_{X}(γ^{p}) = ∞.

Remarks:

• We should notice that the conclusion immediately fails if Γ does not have infinite contortion. Indeed there exists an element γ such that it powers describe only finitely many conjugacy classes of elements g_{1} … g_{q}, and hence

lim sup_{p→∞} d_{X}(γ^{p}) ≤ sup_{i∈{1,...,q}} d_{X}(g_{i}) < ∞.

• It is also interesting to notice that there are groups with infinite contortion which possess elements γ such that

lim inf_{p→∞} d_{X}(γ^{p}) < ∞.

Indeed, there are finitely generated linear groups which contain elements γ which are conjugated to infinitely many of its powers. Hence, for such γ we have

lim inf_{p→∞} d_{X}(γ^{p}) ⩽ d_{X}(γ).

Here is a simple example. We take Γ = SL(2, ℤ[\frac{1}{p}]) and

γ = ( \begin{matrix}1 & 1 \\ 0 & 1 \end{matrix} ) .

Then for all n, γ^{p^{n}} is conjugated to γ.

• However, in Paragraph 5.4, we shall give a condition - bounded depth roots (satisfied, for example, by any group commensurable to a subgroup of SL(n, ℤ)) so that together with the hypothesis of previous Lemma

lim_{p→∞} d_{X}(γ^{p}) = ∞.

Proof : We want to prove that

lim sup_{p→∞} inf_{x∈X} d(x, γ^{p}x) = ∞.

Assume the contrary, then there exists

• a constant R,

• a sequence of points x_{i} of points in X,

such that for every p,

d(x_{i}, γ^{p}x_{i}) ⩽ R.

Let now K be a compact in X such that Γ.K = X. Let f_{i} ∈ Γ such that y_{i} = f_{i}^{−1}(x_{i}) ∈ K. Then

d(y_{i}, f_{i}^{−1}γ^{p}f_{i}(y_{i})) ⩽ R.

Let K_{R} = {z ∈ X, d(z, K) ⩽ R}. It follows that

∀p, f_{i}^{−1}γ^{p}f_{i}(K_{R}) ∩ K_{R} ≠ 0.

Observe that K_{R} is compact. By the properness of the action of Γ, we conclude that the family {g_{i}^{−1}γ_{i}^{p}g_{i}} is finite. Hence the family of conjugacy classes of the sequence γ^{p} is finite. But this contradicts infinite contortion for Γ. Q.e.d.

Corollary 5.2.2 Assume Γ has infinite contortion. Let C be its Cayley graph, then for γ a non torsion element

lim sup_{p→∞} ℓ(γ^{p}) = ∞.

Corollary 5.2.3 Assume Γ has infinite contortion. Let ρ be a representation of dimension n. Assume ρ(Γ) contains a non trivial unipotent, then ρ is not well displacing on X_{n} = SL(n, ℝ)/SO(n, ℝ).

Proof : Assume ρ is well displacing. Let γ such that ρ(γ) is a non trivial unipotent. Then for all p, d_{X_{n}}(γ^{p}) = 0. However γ is not a torsion element. We obtain the contradiction using the previous lemma. Q.e.d.

5.3 Non uniform lattices

Lemma 5.3.1 For n ⩾ 3, the action of SL(n, ℤ) on X_{n} is such that the orbit maps are quasi isometric embeddings. But it is not well displacing.

Proof : The group SL(n, ℤ) is residually finite. Hence it has infinite contortion by Lemma 5.1.1. The standard representation ρ contains a non trivial unipotent hence it is not well displacing by Corollary 5.2.3.

By a theorem of Lubotzky, Mozes and Ragunathan [8] irreducible higher rank lattices Λ are quasi-isometrically embedded in the symmetric space. Q.e.d.

5.4 Bounded depth roots

This section is complementary. We say a group Γ has bounded depth roots property, if for every γ in Γ which is a non torsion element, there exists some integer p, such that we have

q ⩾ p, η ∈ Γ ⟹ η^{q} ≠ γ.

Observe that SL(2, ℤ[\frac{1}{p}]) does not have bounded depth roots. Note that this property is well behaved by taking subgroups and is a property of commensurability (see Lemma 5.4.4).

We prove

Proposition 5.4.1 The group SL(n, ℤ) has bounded depth roots property. Groups of the form G(O) = G(F) ∩ SL(n, O) have bounded depth roots property where O is the ring of integers of a number field F and G ⊂ SL(n) is defined over F. In particular the arithmetic lattices arising in this way have bounded depth roots property.

Lemma 5.4.2 Bounded depth root Let Γ be a group with bounded depth roots property. Assume Γ acts cocompactly and properly discontinuously by isometry on a space X. Assume furthermore that every closed bounded set in X is compact. Then, for every non torsion element γ in Γ, we have

lim_{p→∞} d_{X}(γ^{p}) = ∞.

For the proof see Pargaraph 5.4.3. The Lemma and Proposition above again imply that the action of SL(n, ℤ) on X_{n} is not well displacing.

5.4.1 Bounded depth roots property for SL(n, ℤ)

We prove the above Proposition.

Lemma 5.4.3 SL(n, ℤ) has bounded depth roots.

Proof : Let A ∈ SL(n, ℤ). Let B ∈ SL(n, ℤ). We assume there exists k such that B^{k} = A. Let {λ_{j}^{A}} and {λ_{j}^{B}} be the eigenvalues of A and B respectively. Let

K = sup_{j} |λ_{j}^{A}|.

Then,

sup_{j} |λ_{j}^{B}| ⩽ K^{\frac{1}{k}} ⩽ K.

Hence all the coefficients of the characteristic polynomial of B have a bound K_{1} which only depends on A. Therefore, since these coefficients only take values in ℤ, it follows the characteristic polynomials of B belongs to the finite family

P = {P (x) = x^{n} + ∑_{k=0}^{k=n−1} a_{k}x^{k} : a_{k} ∈ ℤ, |a_{k}| ⩽ K_{1}}

Since P is finite, there exists a constant b > 1 such that for every root λ of a polynomial P ∈ P,

|λ| > 1 ⟹ |λ| ⩾ b.

Let q ∈ N be such that b^{q} is greater than K. It follows that if B^{q} = A, then all eigenvalues of B have complex modulus 1. Therefore the same holds for A.

It follows from this discussion that we can reduce to the case where

∀i, j, |λ_{j}^{A}| = 1.

We say such an element has trivial hyperbolic part. Note that necessarily also B has trivial hyperbolic part.

We first prove that there exist a integer M depending only on n, such that if C ∈ SL(n, ℤ) has a trivial hyperbolic part then C^{M} is unipotent. The same argument as above show that the characteristic polynomials of elements with a trivial hyperbolic part belong a finite family of the form

P = {P (x) = x^{n} + ∑_{k=0}^{k=n−1} a_{k}x^{k} : a_{k} ∈ ℤ, |a_{k}| ⩽ K_{2}}.

Where K_{2} depends only on n. Note that roots of polynomials belonging to P that are of complex modulus 1 are roots of unity. Thus we may take M to be a common multiple of the orders of those roots of unity and deduce that C^{M} is unipotent if C ∈ SL(n, ℤ) has a trivial hyperbolic part.

Returning to our setting we can replace A by A^{M} and B by B^{M} and consider A = B^{k} where both A and B are unipotents. There is some rational matrix g_{0} ∈ SL(n, ℚ) depending on A such that A_{0} = g_{0}Ag_{0}^{−1} is in a Jordan form. We claim that B_{0} = g_{0}Bg_{0}^{−1} is a made of blocks which correspond to the Jordan blocks of A_{0} and each such block of B_{0} is an upper triangular unipotent matrix (This follows from observing that for unipotent matrices a matrix and its powers have the same invariant subspaces). Moreover note that the denominators of the entries of B_{0} are bounded by some L ∈ ℕ depending only on g_{0} (and thus on A). By considering (some of the) entries just above the main diagonal it is easily seen that k ≤ L. Q.e.d.

5.4.2 Commensurability

We observe

Lemma 5.4.4 If Γ is commensurable to a subgroup of a group which has bounded depth roots. Then Γ has bounded depth root.

Proof : By definition a subgroup of a group having bounded depth roots has bounded depth roots. Let G be a group and H a subgroup having finite index k. Observe that for every element g of G, we have g^{k} ∈ H. It follows that if H has bounded depth roots, then G has bounded depth roots. Q.e.d.

5.4.3 Proof of Lemma 5.4.2.

Proof : Let K be a compact in X. We first prove that

lim_{p→∞} inf_{x∈K,η∈Γ} d(x, η^{−1}γ^{p}ηx) = ∞.

Assume the contrary, then there exists

• a constant R,

• a sequence of integers p_{i} going to infinity,

• a sequence of points x_{i} in K,

• a sequence of elements η_{i} of Γ,

such that d(x_{i}, η_{i}^{−1}γ^{p_{i}}η_{i}x_{i}) ⩽ R. It follows that

∀i, (η_{i}^{−1}γη_{i})^{p_{i}}K_{R} ∩ K_{R} ≠ 0.

By the properness of the action of Γ, we conclude that the family {(η_{i}^{−1}γη_{i})^{p_{i}}} is finite. But this contradicts the bounded depth root property.

We now choose the compact K such that Γ.K = X. It follows that

lim_{p→∞} inf_{x∈X} d(x, γ^{p}x) = lim_{p→∞} inf_{x∈K,η∈Γ} d(ηx, γ^{p}ηx) = ∞.

This is what we wanted to prove. Q.e.d.

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