Marc Bourdon
email: bourdon@math.univ-lille1.fr.

Mostow type rigidity theorems

Keywords: 2010 Mathematics Subject Classification: 20F65, 20F67, 20F69, 22E40, 30C65, 30L10 Hyperbolic groups and nonpositively curved groups, asymptotic properties of groups, discrete subgroups of Lie groups, quasiconformal mappings

teaching and research institutions in France or abroad, or from public or private research centers.

Introduction

The Mostow celebrated rigidity theorem for rank-one symmetric spaces states that every isomorphism between fundamental groups of compact, negatively curved, locally symmetric manifolds, of dimension at least 3, is induced by an isometry. In his proof, Mostow exploits two major ideas: group actions on boundaries and regularity properties of quasi-conformal homeomorphisms. This set of ideas revealed itself very fruitful. It forms one of the bases of the theory of Gromov hyperbolic groups. It also serves as a motivation to develop quasi-conformal geometry on metric spaces.

The present text attempts to provide a synthetic presentation of the rigidity theorems that rely on the quasi-conformal geometry of boundaries of hyperbolic spaces. Previous surveys on the subject include [START_REF] Gromov | Rigidity of lattices: an introduction[END_REF][START_REF] Bourdon | Quasi-conformal geometry and hyperbolic geometry[END_REF][START_REF] Bonk | Quasiconformal geometry of fractals[END_REF][START_REF] Kleiner | The asymptotic geometry of negatively curved spaces: uniformization, geometrization and rigidity[END_REF][START_REF] Haissinsky | Géométrie quasiconforme, analyse au bord des espaces métriques hyperboliques et rigidités[END_REF]. The originality of this text lies more in its form. It has two objectives. The first one is to discuss and prove some classical results like Mostow's rigidity in rank one, Ferrand's solution of Lichnérowicz's conjecture, the Sullivan-Tukia and the Pansu quasi-isometry rigidity theorems. The second one is to present more briefly some of the numerous recent advances and results based on the quasi-conformal geometry of boundaries.

The paper starts with three preliminary sections. Section 2 is a survey on Gromov hyperbolic spaces, their boundaries and CAT(-1)-spaces. Section 3 concerns rank-one non-compact symmetric spaces. We describe in detail their boundaries in relation to the nilpotent structure. Section 4 presents the geometric analytic tools that serve in the sequel. This includes the convergence property, the Sullivan characterization of Möbius homeomorphisms, the notion of Loewner spaces and the regularity properties of quasi-conformal homeomorphisms. The heart of the paper is formed by Sections 5,[START_REF] Besson | Entropies et rigidités des espaces localement symétriques de courbure strictement négative[END_REF][START_REF] Besson | Minimal entropy and Mostow's rigidity theorems[END_REF][START_REF] Bestvina | The boundary of negatively curved groups[END_REF]. Each of them consists of a main part with major statements and sketches of proofs, and of a succinct exposition of several related results. Section 5 is devoted to the Mostow original theorem and its proof. We also state some generalizations, including Besson-Courtois-Gallot's theorem. In Section 6, Ferrand's theorem and Sullivan-Tukia's theorem are stated and proved by using the zoom method. We also present Tukia's proof of Mostow's theorem (for real hyperbolic spaces). Section 7 focuses on rigidity of quasi-isometries. A detailed sketch of the proof of Pansu's theorem is given. Right-angled Fuchsian buildings are also discussed. Finally, Section 8 presents some recent developments and perspectives, including Cannon's conjecture and the combinatorial Loewner property.

Clearly, several results presented in this paper would deserve a more detailed exposition. Moreover only very few aspects of rigidity are treated. In particular, infinitesimal rigidity, higher rank rigidity, superrigidity, harmonic maps and bounded cohomology, do not appear at all. This is due to my own limitations.

It is my pleasure to refer to P. Haïssinsky's survey [START_REF] Haissinsky | Quasi-Moebius group actions[END_REF], in this volume, for a complementary viewpoint of some of the subjects discussed here.

is a homothety if it is an isometry after possibly rescaling the metric of X or Y by a multiplicative positive constant.

Two real-valued functions f, g defined on a space X are said to be comparable, and we write f g, if there exists a constant C > 0 such that C -1 f ≤ g ≤ Cf . We write f g if there is a constant C > 0 such that f ≤ Cg.

Gromov hyperbolic spaces

Hyperbolic spaces and hyperbolic groups were defined by Gromov [START_REF] Gromov | Hyperbolic groups[END_REF] in the middle of the eighties. We give a brief survey on hyperbolic spaces and their boundaries, see e.g. [START_REF] Gromov | Hyperbolic groups[END_REF][START_REF] Coornaert | Géométrie et théorie des groupes, les groupes hyperboliques de Gromov[END_REF][START_REF]Sur les groupes hyperboliques d'après Mikhael Gromov[END_REF][START_REF] Bridson | Metric spaces of non-positive curvature[END_REF][START_REF] Kapovich | Boundaries of hyperbolic groups[END_REF][START_REF] Buyalo | Elements of asymptotic geometry[END_REF][START_REF] Drutu | Lectures on Geometric Group Theory[END_REF] for more details.

First definitions and properties

Let (X, d) be a metric space. A geodesic segment in X is an isometrically embedded closed interval. A geodesic segment with endpoints x, y ∈ X is denoted by [x, y]. Similarly are defined the geodesic rays and the bi-infinite geodesics in X. The space X is geodesic if every pair of points in X can be joined by a geodesic segment. Definition 2.1. A geodesic metric space X is hyperbolic if there is a constant δ ≥ 0 such that for every points x, y, z ∈ X and every geodesic segments [x, y], [y, z],

[z, x], one has [x, y] ⊂ N δ ([yz] ∪ [zx]).
A companion definition is : Definition 2.2.

1. A map F : X 1 → X 2 between metric spaces is a quasi-isometric embedding, if there are constants λ ≥ 1 and C ≥ 0 such that for every x, y ∈ X 1 λ -1 d(x, y) -C ≤ d(F (x), F (y)) ≤ λd(x, y) + C.

In the particular case where X 1 is an interval in R, the map F is called a (λ, C)-quasi-geodesic of X 2 .

2. The map F is a quasi-isometry if it is a quasi-isometric embedding and if there is a constant R ≥ 0 such that N R (F (X 1 )) = X 2 .

Examples are provided by the following situation. A group action is called geometric if it is an isometric, properly discontinuous and cocompact action. The Cayley graph of a finitely generated group (Γ, S), is the graph whose vertices are the elements of Γ, and whose edges are the pairs {g, gs}, with g ∈ Γ and s ∈ S ∪ S -1 . It is endowed with the path metric obtained by identifying every edge with the unit interval.

Proposition 2.3 (Svarc-Milnor). Suppose a group Γ acts geometrically on a geodesic metric space X. Let S be a finite generating set of Γ. Then for every O ∈ X, the orbit map Γ → X, g → g • O, extends to a quasi-isometry of the Cayley graph of (Γ, S) to X.

The following result is called the Morse Lemma in [START_REF] Gromov | Hyperbolic groups[END_REF].

Theorem 2.4. Suppose X is a δ-hyperbolic space. There is a constant R = R(λ, C, δ), such that for every (λ, C)-quasi-geodesic γ : I → X there exists a geodesic c ⊂ X with γ(I) ⊂ N R (c).

As a consequence one obtains Corollary 2.5. Suppose F : X 1 → X 2 is a quasi-isometry between geodesic metric spaces. Then X 1 is hyperbolic if and only if X 2 is.

Since the Cayley graphs of Γ, for the various choices of generators, are pairwise quasi-isometric, the following definition does not depend on the finite generating set S. Definition 2.6. A finitely generated group Γ is called hyperbolic if the Cayley graph of (Γ, S) is hyperbolic.

Boundary at infinity

The boundary at infinity of a hyperbolic space, equipped with a visual metric, is the main protagonist of the paper. The boundary at infinity is defined in Definition 2.7, the visual metrics are defined in the statement of Theorem 2.9. Definition 2.7. Let X be a geodesic metric space. Geodesic rays r 1 , r 2 : [0+∞) → X are asymptotic if sup t∈[0,+∞) d(r 1 (t), r 2 (t)) < ∞.

The boundary at infinity of X is

∂X := {r : [0 + ∞) → X geodesic ray}/ ∼
where r 1 ∼ r 2 when they are asymptotic. It is endowed with the topology induced by the topology of uniform convergence on the compact subsets of [0, +∞). The group Isom(X) acts on ∂X by homeomorphisms.

Recall that a metric space is proper if its closed balls are compact. Using Ascoli's theorem one has : Proposition 2.8. Suppose that X is a proper hyperbolic space.

1. Given an origin O ∈ X, every geodesic ray of X is asymptotic to a geodesic ray starting at O.

2.

For every pair of non-asymptotic geodesic rays r 1 , r 2 , there is a geodesic γ : R → X that is asymptotic to r 1 for t ≤ 0 and to r 2 for t ≥ 0.

3. The boundary ∂X is compact.

In the sequel, we denote by (z, w) every geodesic which is asymptotic to a pair of distinct points z, w ∈ ∂X. Note that (z, w) is not unique in general. But for two of such geodesics γ, η one has γ ⊂ N 10δ (η), thanks to δ-hyperbolicity. Theorem 2.9 (Gromov). Suppose that X is a proper δ-hyperbolic space, and let O ∈ X be an origin. Then:

1. There is a constant a 0 > 1 which depends only on δ, such that for every a ∈ (1, a 0 ), there exists a metric d on ∂X with the following property: for every z, w ∈ ∂X, one has d(z, w) a -L , where L = dist(O, (z, w)).

A metric d on ∂X which satisfies the above property is called a visual metric.

2. X ∪ ∂X is a natural metric compactification of X. More precisely, there is a metric d on X ∪ ∂X that enjoys the following property: for every x, y ∈ X ∪ ∂X, one has

d(x, y) a -L min{1, |x -y|}, where L = dist(O, (x, y)).
Observe that two visual metrics d, δ on ∂X, with parameters a and b respectively (and with possibly different origins) satisfy δ d log a log b . Sketch of proof. We sketch a proof of Part 1, and refer to [START_REF] Bonk | Uniformizing Gromov hyperbolic spaces[END_REF] for Part 2. Suppose first that X is a tree. Then for every distinct z, w ∈ ∂X the geodesic (z, w) is unique, and for every a > 1 the fonction

d(z, w) := a -dist(O,(z,w))
satisfies the ultrametric inequality

d(z 1 , z 3 ) ≤ max{d(z 1 , z 2 ), d(z 2 , z 3 )}.
For a general δ-hyperbolic space X, the fonction

ϕ : (z, w) ∈ (∂X) 2 → e -dist(O,(z,w)) satisfies ϕ(z 1 , z 3 ) ≤ e 100δ max{ϕ(z 1 , z 2 ), ϕ(z 2 , z 3 )}.
Therefore part 1 of the theorem is a consequence of the following classical lemma (see e.g. [START_REF] Heinonen | Lectures on analysis on metric spaces[END_REF] for a proof).

Lemma 2.10. Let Z be a set and ϕ : Z × Z → [0, +∞) be a quasi-metric, i.e. a function that satisfies the following properties:

1. ϕ(z, w) = 0 if and only if z = w.

2. ϕ(w, z) = ϕ(z, w) for every w, z ∈ Z.

3. There is a K > 0 such that for every z 1 , z 2 , z 3 ∈ Z one has

ϕ(z 1 , z 3 ) ≤ K max{ϕ(z 1 , z 2 ), ϕ(z 2 , z 3 )}.
Then there exists α 0 ∈ (0, 1) which depends only on K, such that for every α ∈ (0, α 0 ) there exists a metric d on Z with d ϕ α .

Cross-ratio and boundary extensions of quasi-isometries

In classical hyperbolic geometry, isometries extend to Möbius homeomorphisms of the boundary. This property generalizes, in some sense, to quasi-isometries of Gromov hyperbolic spaces.

Definition 2.11 (Väisälä [START_REF]Quasi-möbius maps[END_REF]). Let (Z, d) be a metric space. The cross-ratio of four pairwise distinct points

z 1 , z 2 , z 3 , z 4 ∈ Z is [z 1 , z 2 , z 3 , z 4 ] = d(z 1 , z 3 )d(z 2 , z 4 ) d(z 1 , z 4 )d(z 2 , z 3 ) . A map f : Z 1 → Z 2 between metric spaces is Möbius if it preserves the cross-ratio. It is quasi-Möbius if there is a homeomorphism η : [0, +∞) → [0, +∞) such that for every four pairwise distinct points z 1 , z 2 , z 3 , z 4 ∈ Z 1 , one has [f (z 1 ), f (z 2 ), f (z 3 ), f (z 4 )] ≤ η([z 1 , z 2 , z 3 , z 4 ]).
The group of Möbius homeomorphisms of Z is denoted by Moeb(Z).

Note that switching z 1 and z 2 in the last inequality leads to the opposite inequality (with a different fonction η). Therefore inverses and compositions of quasi-Möbius homeomorphisms are quasi-Möbius as well.

Theorem 2.12 (Efremovich-Tihomirova [START_REF] Efremovich | Equimorphisms of hyperbolic spaces[END_REF]). Let X 1 and X 2 be proper δ-hyperbolic spaces. Equip ∂X 1 and ∂X 2 with visual metrics. Then every quasi-isometry F : X 1 → X 2 extends to a quasi-Möbius homeomorphism f : ∂X 1 → ∂X 2 . Moreover the distortion fonction η depends only on δ, the constants of the quasiisometry F and the constants of the visual metrics.

In particular Isom(X) acts on ∂X by uniform quasi-Möbius homeomorphisms, i.e. with the same distortion function.

Sketch of proof.

Let O be an origin in X 1 . In order to extend the quasi-isometry F to a map f : ∂X 1 → ∂X 2 , consider z ∈ ∂X 1 and a geodesic ray [O, z). Its image by F is a quasi-geodesic ray. By the Morse lemma it lies within bounded distance from a geodesic ray [F (O), w), with w ∈ ∂X 2 . Define f (z) = w. It is easy to see that f is bijective.

To see that f is quasi-Möbius, one observes that there is a constant C ≥ 0 which depends only on δ and the constants of the visual metric, such that for every pairwise distinct points z 1 , z 2 , z 3 , z 4 ∈ ∂X, one has

| dist ((z 1 , z 4 ), (z 2 , z 3 )) -max 0, log[z 1 , z 2 , z 3 , z 4 ] log a | ≤ C.
These inequalities follow easily from the special case where X is a tree (in this case C = 0). By combining the above inequalities with the Morse lemma one obtains that f is quasi-Möbius.

We note that the converse of Theorem 2.12 is valid for non-degenerate hyperbolic spaces. A hyperbolic space X is non-degenerate if there is a constant C ≥ 0 such that every x ∈ X lies within distance at most C from all three sides of some ideal geodesic triangle ∆ x . According to [START_REF] Paulin | Un groupe hyperbolique est déterminé par son bord[END_REF][START_REF] Bonk | Embeddings of Gromov hyperbolic spaces[END_REF], every quasi-Möbius homeomorphism f : ∂X 1 → ∂X 2 , between boundaries of proper non-degenerate hyperbolic spaces, extends to a quasi-isometry F : X 1 → X 2 . Therefore the boundary is a full quasi-isometric invariant of the proper non-degenerate hyperbolic spaces.

Some dynamical properties

We give some properties of ∂X for the hyperbolic spaces X that admit a geometric action. The definition below of approximately self-similar spaces appears in [START_REF] Kleiner | The asymptotic geometry of negatively curved spaces: uniformization, geometrization and rigidity[END_REF]. Definition 2.13.

1. A metric space Z is approximately self-similar if there is a constant L 0 ≥ 1 such that if B(z, r) ⊂ Z is an open ball of radius 0 < r ≤ diam(Z), then there is an open subset U ⊂ Z which is L 0 -bi-Lipschitz homeomorphic to the rescaled ball (B(z, r), 1 r d). 2. A metric space Z is Ahlfors Q-regular (for some Q ∈ (0, +∞)) if there is a measure ν on Z such that for every ball B ⊂ Z of radius 0 < r ≤ diam(Z) one has ν(B) r Q .
Note that if Z is Ahlfors Q-regular, then Q is its Hausdorff dimension. Moreover the measure ν and the Q-Hausdorff measure are absolutely continuous with respect to each other, and their Radon-Nikodym derivatives are bounded. Observe also that an approximately self-similar space is Ahlfors Q-regular as soon as its Q-Hausdorff measure is finite and non-zero.

Theorem 2.14. Let X be a proper hyperbolic space such that |∂X| ≥ 3, and suppose that a group Γ acts on X geometrically. Let d be a visual metric on ∂X, let a be its parameter, and let O ∈ X be an origin. Then :

1. (∂X, d) is approximately self-similar, the partial bi-Lipschitz maps being restrictions of elements of Γ.

(∂X, d) is Ahlfors

Q-regular with Q = lim sup R→+∞ log |(Γ • O) ∩ B(O, R)| R log a .
3. Let H be the Q-Hausdorff measure of (∂X, d). The diagonal action of Γ on

((∂X) 2 , H × H) is ergodic.
Statement 2 is due to M. Coornaert [START_REF] Coornaert | Sur les groupes proprement discontinus d'isométries des espaces hyperboliques de Gromov[END_REF][START_REF] Coornaert | Mesures de Patterson-Sullivan sur le bord d'un espace hyperbolique au sens de Gromov[END_REF]. Statement 3 is proved in [START_REF] Bader | Some ergodic properties of metrics on hyperbolic groups[END_REF]. They generalize previous results of Patterson and Sullivan to hyperbolic spaces and groups. See e.g. [START_REF] Bourdon | Combinatorial modulus, the combinatorial Loewner property, and Coxeter groups[END_REF] for a proof of 1.

M. Bourdon

Let again X be a proper hyperbolic space X, suppose that |∂X| ≥ 3, and consider the topological space

∂ 3 X := {(z 1 , z 2 , z 3 ) ∈ (∂X) 3 | z i = z j for i = j},
with the diagonal action of Isom(X) by homeomorphisms.

Proposition 2.15. Suppose that a group Γ acts on X geometrically. Then Γ acts properly discontinuously and cocompactly on ∂ 3 X.

Conversely, suppose that X is a proper non-degenerate hyperbolic space (as defined in the last subsection). Then every group Γ ⊂ Isom(X) that acts properly discontinuously and cocompactly on ∂ 3 X, acts geometrically on X.

Sketch of proof. To every (z 1 , z 2 , z 3 ) ∈ ∂ 3 X, one associates an ideal triangle in X, i.e. the union of three bi-infinite geodesics (z i , z j ), i = j, denoted by ∆(z 1 , z 2 , z 3 ). Then, by hyperbolicity, one constructs a center of the triangle ∆(z 1 , z 2 , z 3 ), i.e. a point x ∈ X such that the maximal of its distances to the three sides is minimal. This defines a map p : ∂ 3 X → X, which is essentially proper and Isom(X)equivariant (see [START_REF] Coornaert | Sur les groupes proprement discontinus d'isométries des espaces hyperboliques de Gromov[END_REF] for more details).

CAT(-1)-spaces

A vast class of hyperbolic spaces is formed by the so-called CAT(-1)-spaces. We recall their definition and review some properties of their boundary. Definition 2.16. Let H 2 R be the real hyperbolic plane of curvature -1. Let X be a geodesic metric space, and let ∆ ⊂ X be a geodesic triangle. A comparison triangle of ∆ is a geodesic triangle ∆ ⊂ H 2 R whose edge lengths are the same as in ∆. Let s : ∆ → ∆ be the natural map. Then X is a CAT(-1)-space if for every geodesic triangle ∆ ⊂ X, the map s : ∆ → ∆ is 1-Lipschitz.

Thanks to the Alexandrov comparison theorem, any simply connected complete Riemannian manifold of sectional curvatures less than or equal to -1 is a CAT(-1)-space. Other examples include the negatively curved simply connected polyhedron complexes [START_REF] Ballmann | Singular spaces of nonpositive curvature[END_REF][START_REF] Bridson | Metric spaces of non-positive curvature[END_REF], and among them Tits buildings associated to hyperbolic Coxeter groups [START_REF] Davis | The geometry and topology of Coxeter groups[END_REF].

A nice feature about CAT(-1)-spaces is the existence of canonical explicit visual metrics on their boundary and the relations between the groups Moeb(∂X) and Isom(X). To present them, we start with some definitions.

Let X be a proper CAT(-1)-space and let O ∈ X be an origin. Denote by |x -y| the distance between x, y ∈ X. Since X is hyperbolic, its boundary appears as the frontier of the compactification X ∪∂X (Theorem 2.9.2). By using Definition 2.16 and some standard geometric properties of H 2 R , one sees that the function

(x, y) ∈ X 2 → 1 2 (|O -x| + |O -y| -|x -y|) ∈ R,
extends by continuity to a function on ∂ 2 X := {(z, w) ∈ (∂X) 2 | z = w}, denoted by (z|w) and called the Gromov product of z, w. Similarly, the function

(x 1 , x 2 , x 3 , x 4 ) ∈ X 4 → 1 2 (|x 1 -x 3 | + |x 2 -x 4 | -|x 1 -x 4 | -|x 2 -x 3 |) ∈ R,
extends by continuity to a function on

∂ 4 X := {(z 1 , z 2 , z 3 , z 4 ) ∈ (∂X) 4 | z i = z j for i = j}, denoted by (z 1 |z 2 |z 3 |z 4 ).
Proposition 2.17 ([26]). Let X be a proper CAT(-1)-space. Then

d CAT (z, w) = e -(z|w)
defines a visual metric of parameter e on ∂X. As a consequence of Proposition 2.17, every isometry of X acts on ∂X as a Möbius homeomorphism with respect to the cross-ratio associated to the visual metric d CAT . A basic inverse problem is the reconstruction of X from the boundary cross-ratio. The following result is an example of such a reconstruction. Theorem 2. 19 ([27]). Let S be a rank-one non-compact symmetric space, and let X be a CAT(-1)-space. Assume the Riemannian metric of S is normalized so that the maximum of the sectional curvatures is equal to -1. Then every embedding ∂S → ∂X, which preserves the cross-ratio, extends to a totally geodesic isometric embedding S → X. In particular Isom(S) = Moeb(∂S).

The idea of proof is the following. Every geodesic in S or X is determined by its two end-points in the boundary. The map ∂S → ∂X induces an injection from the set of geodesics in S to the set of geodesics in X. By using the metric d CAT one can show that concurrent geodesics are preserved.

Rank-one symmetric spaces

Examples of CAT(-1)-spaces include the rank-one non-compact symmetric spaces. These are the hyperbolic spaces

H n R , H n C , H n Q , H 2 O ,
where n ≥ 2 and where Q and O denote the quaternions and the octonions, respectively. In the sequel K denotes one of the fields R, C, Q, and we set k := dim R K.

This section gives a brief presentation the rank-one (non-compact) symmetric spaces and discusses some aspects of their boundaries. We start with a definition of the hyperboloid and parabolic models of H n K . Then we describe the nilpotent structure on ∂H n K \ {∞}, and we compare the visual metric d CAT with the Carnot-Carathéodory metric on ∂H n K . Althrough the material is standard -see [START_REF] Mostow | Strong rigidity of locally symmetric spaces[END_REF][START_REF] Pansu | Métriques de Carnot-Carathéodory et quasiisométries des espaces symétriques de rang un[END_REF][START_REF] Gromov | Rigidity of lattices: an introduction[END_REF][START_REF] Bridson | Metric spaces of non-positive curvature[END_REF][START_REF] Mackay | Conformal dimension: theory and application[END_REF] -we provide a detailed and self-contained exposition.

Because O is non-associative, the space H 2 O requires a different treatment. For simplicity we consider only H n R , H n C , H n Q , and we refer to [START_REF] Mostow | Strong rigidity of locally symmetric spaces[END_REF][START_REF] Allcock | Reflections groups on the octave hyperbolic plane[END_REF] for the octonionic case.

Hyperboloid model

Let n ≥ 1. We consider K n+1 as a right K-module (1 ). Let B :

K n+1 × K n+1 → K be the form B(x, y) = x 1 y 1 + ... + x n y n -x n+1 y n+1 ,
where x → x denotes the standard involution of K. The associated quadratic form is q(x)

:= B(x, x) = |x 1 | 2 + ... + |x n | 2 -|x n+1 | 2 .
Let H = {x ∈ K n+1 ; q(x) = -1}. The group U := {a ∈ K ; |a| = 1} acts on H on the right, and preserves q. Moreover, when restricted to every orthogonal tangent space of the U -orbits, q is positive definite.

The hyperboloid model of H n K is the manifold H/U equipped with the Riemannian metric induced by q.

Observe that H/U lies in the projective space P n K . Indeed, it is the image of H by the projection map π : K n+1 \ {0} → P n K . The description of the Riemannian metric is the following. Proposition 3.1. Denote by g the Riemannian metric on H n K . Then for x ∈ H and v ∈ T x H one has:

g π(x) (π * (v), π * (v)) = q(v) + |B(x, v)| 2 .
Proof. The tangent space to xU at x is x K. Its orthogonal in T x H is the set of v ∈ K n+1 such that B(x, v) = 0. Since x ∈ H and v ∈ T x H, one has q(x) = -1 and B(x, v) ∈ K. Thus the component of v along the orthogonal of xU is w = v + xB(x, v). We obtain

g π(x) (π * (v), π * (v)) = q(w) = B(w, w) = q(v) + |B(x, v)| 2 ,
as expected.

In this model ∂H n

K is identified with the submanifold {q(x) = 0}/K * of P n K . The equation B(x, v) = 0 defines a codimension k -1 distribution in the tangent space of {q(x) = 0}. Its leaves contain the tangent spaces of the K * -orbits. Therefore it induces in the quotient, a codimension k -1 distribution on ∂H n K that we denote by T . We remark that T is the maximal K-subbundle of the tangent bundle of ∂H n K .

The groups P O(n, 1), P U (n, 1), P Sp(n, 1) act by isometries on

H n R , H n C , H n Q
respectively, and they preserve T on the boundary. They are in fact equal to the whole isometry groups of the corresponding symmetric spaces.

Parabolic model

We now describe a model of H n K where the point [0, ..., 0, -1, 1] ∈ ∂H n K lies at infinity. It is called the parabolic model or Siegel domain.

We consider the K-hyperplane P in K n+1 defined by x n +x n+1 = 1. It contains the point (0, ..., 0, 1) and is parallel to the vector (0, ..., 0, -1, 1). We introduce the following coordinates on P and P n K :

(x , x n ) ∈ K n → (x , 1 2 -x n , 1 2 + x n ) ∈ P → [x , 1 2 -x n , 1 2 + x n ] ∈ P n K .
In these coordinates, the quadratic form q restricted to P is expressed as follows.

For (x , x n ) ∈ K n : q(x , x n ) = x 2 -2 x n , with x 2 := |x 1 | 2 + ... + |x n-1 | 2 .
Therefore, by setting ∞ := [0, ..., 0, -1, 1], one gets the following expressions

H n K = {(x , x n ) ∈ K n ; x 2 -2 x n < 0}, ∂H n K \ {∞} = {(x , x n ) ∈ K n ; x 2 -2 x n = 0}.
A computation based on Proposition 3.1 shows that the Riemannian metric is expressed as follows. For

x = (x , x n ) ∈ H n K and v = (v , v n ) ∈ K n : g x (v, v) = v 2 2 x n -x 2 + |{x , v } -v n | 2 x n -x 2 2 , (3.2)
where {•, •} denotes the form {x , y } = x 1 y 1 +...+x n-1 y n-1 . Another computation shows that the distribution T on ∂H n K \ {∞} is simply written as

{x , v } = v n . (3.3) 3.3 Nilpotent structure on ∂H n K \ {∞}
We use the coordinate system of the parabolic model. Let N be the Lie group K n-1 × K with the following multiplication law

(z , z n ) • (w , w n ) = (z + w , z n + w n + {z , w }).
Its Lie algebra is written as n = K n-1 ⊕ K, with K central, and for every z , w ∈ K n-1 : [z , w ] = {z , w }.

(3.4)

Therefore when K = R the group N is two-step nilpotent and otherwise it is abelian. It acts on

H n K as follows. For (z , z n ) ∈ N and (x , x n ) ∈ H n K , (z , z n ) • (x , x n ) = (x + z , x n + z n + {z , x } + 1 2 z 2 ).
By using the expression (3.2), one checks that it is an isometric action. Moreover N acts simply transitively on ∂H n K \ {∞}. Therefore we get an identification

N → ∂H n K \ {∞}, (z , z n ) → (z , z n ) • 0,
In these coordinates, the distribution T is equal to the left invariant distribution on N generated by the subspace K n-1 ⊂ n (this follows from (3.3)). The Lie bracket expression (3.4) implies that

T ⊕ [T, T ] = T (∂H n K ). (3.5)
Let (δ t ) t∈R be the 1-parameter subgroup of Isom(H n K ) defined by

δ t (x , x n ) = (e t x , e 2t x n ).
Its elements are hyperbolic isometries with axis (0, ∞). It normalizes N . More precisely we have for every (z , z n ) ∈ N :

δ t • (z , z n ) • δ -1 t = (e t z , e 2t z n ).
The corresponding Lie algebra automorphisms are given by e tα , where α is the derivation of n, equal to id on K n-1 and to 2id on K.

The map δ t preserves the distribution T . It acts as a similarity of ratio e t on the leaves of T equipped with any N -invariant Riemannian metric. Indeed the distribution T is N -invariant, δ t normalizes N , and its differential at the fixed point 0 is a homothety of ratio e t when restricted to the leaf of T at 0.

The above properties show that N is a Carnot group with associated Carnot homotheties (δ t ) t∈R . The required definitions are the following. Definition 3.6. A Carnot group is a pair (N, v), where N is a simply connected Lie group, and v is a linear subspace of the Lie algebra n such that

n = r≥1 v r , with v 1 := v and v r+1 := [v, v r ].
The linear subspace v is called the horizontal space. The linear map α, whose restriction to every v r is r • id, is a derivation of n. The associated Carnot homotheties are the automorphisms of N induced by e tα ∈ Aut(n).

Carnot-Carathéodory and visual metrics

The sectional curvature of H n K is constant equal to -1 when K = R and lies in [-4, -1] otherwise. Thus H n K is a CAT(-1)-space and its boundary carries the visual metrics d CAT , see Proposition 2.17.

On the other hand, the relation (3.5) shows that the distribution T generates the tangent space of ∂H n K . Thus, according to Chow's theorem, any two boundary points can be joined by a smooth horizontal curve, i.e. whose tangent vectors lie in T . Choose an origin O ∈ H n K , and equip T with a Riemannian metric that is invariant under the isotropy group of O. The associated Carnot-Carathéodory metric d CC is defined as follows. For every z, w ∈ ∂H n K , the distance d CC (z, w) is the infimum of the lengths of the piecewise smooth horizontal curves joining z to w.

∈ ∂H n K , such that d CAT (z 0 , •) d CC (z 0 , •) in a neighborhood of z 0 .
Choose z 0 to be the common fixed point of the Carnot homotheties δ t , see Subsection 3.3. As we saw in this subsection, δ t multiplies by e t the Carnot-Carathéodory metric on ∂H n K \ {∞} associated to any N -invariant Riemannian metric on T . Since this metric is Lipschitz equivalent to d CC in a neighborhood of z 0 , one has for z in a neighborhood of z 0 and t ≤ 0:

d CC (z 0 , δ t (z)) e t d CC (z 0 , z).
On the other hand, the map δ t is a hyperbolic isometry of H n K whose axis contains z 0 and whose translation length is |t|. Thus, with the definition of d CAT , one has for z in a neighborhood of z 0 and t ≤ 0:

d CAT (z 0 , δ t (z)) e t d CAT (z 0 , z).
The expected property comes from these two homogeneity relations.

It remains to compute the Hausdorff dimension. Let d N be the Carnot-Carathéodory metric on N ∂H n K \{∞} associated to an N -invariant Riemannian metric on T . The metrics d N and d CC are locally Lipschitz equivalent, thus their Hausdorff dimensions are equal. The distance d N is N -invariant and is multiplied by e t under δ t . On the other hand, the fonction

φ(z , z n ) := z + |z n | 1/2 (3.8)
is positive on N \ {0} and δ t -homogeneous. Therefore the function Φ(z, w)

:= φ(z -1 w) on N 2 is N -invariant
Hausdim(N, d N ) = dim R K n-1 + 2 dim R K = (n -1)k + 2(k -1) = nk + k -2.
The statement follows.

Some geometric analysis

This section describes some dynamical and analytic tools and results that will serve in the sequel. This includes the convergence property (Proposition 4.1), Sullivan's characterization of Möbius homeomorphisms (Proposition 4.3), some regularity properties of quasi-Möbius homeomorphisms between Loewner spaces (Theorem 4.7), and the relations between quasi-Möbius and quasi-conformal homeomorphisms of Loewner spaces (Theorem 4.10). We refer to the survey [START_REF] Haissinsky | Quasi-Moebius group actions[END_REF] for a complementary viewpoint and more details on some parts of this section.

Convergence property of quasi-Möbius maps

A standard and classical property is the following convergence property of sequences of uniform quasi-Möbius maps. It will serve repeatedly in Section 6.

Proposition 4.1. Let Z 1 , Z 2 be compact metric spaces. Let f k : Z 1 → Z 2 be a sequence of uniform quasi-Möbius homeomorphisms (i.e. we assume the distortion function η to be the same for every f k , k ∈ N). Then one of the following properties occurs:

1. Up to a subsequence, (f k ) k∈N converges uniformly on Z 1 to a η-quasi-Möbius

homeomorphism f : Z 1 → Z 2 .
2. Up to a subsequence, there exist a ∈ Z 1 and b ∈ Z 2 , such that for every com-

pact subset K ⊂ Z \ {a}, (f k ) k∈N converges uniformly on K to the constant map b.
The crucial lemma is the following.

Lemma 4.2. Let f k : Z 1 → Z 2 be a sequence of uniform η-quasi-Möbius maps between compact metric spaces. Suppose there exists three pairwise distinct points a 1 , a 2 , a 3 ∈ Z 1 and three pairwise distinct points b

1 , b 2 , b 3 ∈ Z 2 , such that for i = 1, 2, 3, f k (a i ) → b i when k → +∞.
Then, up to a subsequence, (f k ) k∈N converges uniformly on Z to a η-quasi-Möbius map f :

Z 1 → Z 2 .
Proof. Let η be the common distortion function for the f k 's. We prove that the sequence (f k ) k∈N is equicontinuous, so the statement will follow from Ascoli's theorem. Let D = min{d(a i , a j ) | i = j, i, j = 1, 2, 3}, and let z, w ∈ Z 1 be such that d(z, w) ≤ D/4. Then dist({z, w}, a i ) ≤ D/4 for at most one of the a i 's. Suppose for example that dist({z, w}, {a 2 , a 3 }) > D/4. We have

d(z, w)d(a 2 , a 3 ) d(z, a 3 )d(a 2 , w) d(z, w). Since d(f k (z), b 3 ) and d(b 2 , f k (w)
) are bounded by above by diam(Z 2 ) , we get for every k ∈ N large enough:

d(f k (z), f k (w)) d(f k (z), f k (w))d(b 2 , b 3 ) d(f k (z), b 3 )d(b 2 , f k (w)) d(f k (z), f k (w))d(f k (a 2 ), f k (a 3 )) d(f k (z), f k (a 3 ))d(f k (a 2 ), f k (w)) η d(z, w)d(a 2 , a 3 ) d(z, a 3 )d(a 2 , w) η(Cd(z, w)), for some constant C ≥ 1 independent of k, z, w. Therefore (f k ) k∈N is equicontin- uous.
Proof of the Proposition. Let a 1 , a 2 , a 3 be three distinct points in Z 1 . Up to a subsequence we can assume that for i = 1, 2, 3,

f k (a i ) converges in Z 2 when k → ∞. Let b i be their limits. If
z k = f -1 k (c).
Up to a subsequence we can assume that (z k ) k∈N converges in Z 1 ; let a be its limit. We have a / ∈ {a 1 , a 2 }. Let K ⊂ Z 1 \ {a} be a compact subset. For k ∈ N large enough and w ∈ K and, we have

d(f k (a 1 ), f k (w)) d(f k (a 1 ), f k (a 2 )) d(f k (a 1 ), f k (w))d(f k (z k ), f k (a 2 )) d(f k (a 1 ), f k (a 2 ))d(f k (z k ), f k (w)) η d(a 1 , w)d(z k , a 2 ) d(a 1 , a 2 )d(z k , w) η C d(a, w) , for some constant C ≥ 1 independent of K, k, w. Therefore d(f k (a 1 ), f k (w)) η C d(a, w) • d(f k (a 1 ), f k (a 2 )),
and so f k converges uniformly on K to the constant map b.

Sullivan's characterization of Möbius homeomorphisms

The following result is a main ingredient in Sullivan's ergodic approach to Mostow's rigidity and its variants, that will be presented in Section 5. It will also serve repeatedly in the sequel.

Proposition 4.3 ([128]

). Suppose (Z 1 , d 1 ) and (Z 2 , d 2 ) are Q-regular metric spaces for some Q > 0. Let H 1 , H 2 be the Q-Hausdorff measures of Z 1 and Z 2 respectively. For i = 1, 2 let µ i be the measure

µ i (z, w) = H i (z) × H i (w) d i (z, w) 2Q on Z 2 i . Then a homeomorphism f : Z 1 → Z 2 is Möbius if and only if (f × f ) * µ 2 = Cµ 1 for some constant C > 0. Its proof will use the Lemma 4.4. Let f : (Z 1 , d 1 ) → (Z 2 , d 2 )
be a Möbius homeomorphism between metric spaces with no isolated points. Then 1. For every z ∈ Z 1 the limit

|f (z)| := lim w→z d 2 (f (w), f (z)) d 1 (w, z)
exists and belongs to (0, +∞).

2. For every z, w ∈ Z 1 one has

d 2 (f (z), f (w)) 2 = |f (z)| • |f (w)| • d 1 (z, w) 2 . Proof of Lemma 4.4. Since [f (z 1 ), f (z 2 ), f (w 1 ), f (w 2 )] = [z 1 , z 2 , w 1 , w 2 ], one gets d 2 (f (z 1 ), f (w 1 ))d 2 (f (z 2 ), f (w 2 )) = = d 2 (f (z 1 ), f (w 2 )) d 1 (z 1 , w 2 ) d 2 (f (z 2 ), f (w 1 )) d 1 (z 2 , w 1 ) d 1 (z 1 , w 1 )d 1 (z 2 , w 2 ).
By letting z i → z and w i → w we obtain the second and then the first statement.

Proof of Proposition 4.3. Suppose that f is Möbius. Then, according to Lemma 4.4, we get

f * H 2 = |f | Q H 1 . By substituting in (f × f ) * µ 2 (z, w) = f * H 2 (z) × f * H 2 (w) d 2 (f (z), f (w)) 2Q , it follows from Lemma 4.4.2. that (f × f ) * µ 2 = µ 1 . Conversely if (f ×f ) * µ 2 = Cµ 1 then f * H 2 = ϕH 1 for some measurable function ϕ. Thus (f × f ) * µ 2 (z, w) = ϕ(z)H 2 (z) × ϕ(w)H 2 (w) d 2 (f (z), f (w)) 2Q .
Therefore for almost all (z, w) ∈ Z 2 1 we have

ϕ(z) • ϕ(w) d 2 (f (z), f (w)) 2Q = C d 1 (z, w) 2Q .
By substituting in the cross-ratio expression, one obtains after cancellations that f preserves the cross-ratio almost everywhere in Z 4 . Since the cross-ratio is continuous we get that f is Möbius.

Loewner spaces

Loewner spaces have been introduced by J. Heinonen and P. Koskela [START_REF] Heinonen | Quasiconformal maps in metric spaces with controlled geometry[END_REF]. Most of the classical Euclidean quasi-conformal analysis was generalized to the setting of Loewner spaces. We present only the material that will be useful for us. More discussions on the Loewner spaces can be found in [START_REF] Heinonen | Quasiconformal maps in metric spaces with controlled geometry[END_REF][START_REF] Heinonen | Lectures on analysis on metric spaces[END_REF][START_REF] Cheeger | Differentiability of Lipschitz functions on metric measure spaces[END_REF][START_REF] Heinonen | Sobolev classes of Banach space-valued functions and quasiconformal mappings[END_REF][START_REF] Tyson | Quasiconformality and quasisymmetry in metric measure spaces[END_REF][START_REF] Keith | The Poincaré inequality is an open ended condition[END_REF]. We refer to [START_REF] Heinonen | Lectures on analysis on metric spaces[END_REF] for a gentle introduction to Loewner spaces. We will use the shorthand Loewner space for a metric space that is Q-regular and Q-Loewner for some Q > 1, in the sense of [START_REF] Heinonen | Quasiconformal maps in metric spaces with controlled geometry[END_REF].

Let Z be a Q-regular metric space with Q > 1, and let H be its Hausdorff measure. Let F be a non-void family of continuous curves in Z. Its Q-modulus is defined by

Mod Q (F) = inf ρ Z (ρ) Q dH,
where the infimum is over all F-admissible functions, i.e. measurable functions ρ : Z → [0, +∞] which satisfy γ ρ ≥ 1 for every rectifiable curve γ ∈ F. If there is no rectifiable curve in F, we set Mod Q (F) = 0. The modulus is an outer measure on the full set of continuous curves in Z. Moreover if every curve in

F 2 contains a curve in F 1 one has Mod Q (F 2 ) ≤ Mod Q (F 1 ).
We denote by F(A, B) the family of curves joining two subsets A and B of Z and by Mod

Q (A, B) its Q-modulus. When A ⊂ B(z, r) and B ⊂ Z \ B(z, R) with 0 < 2r < R, then one has Mod Q (A, B) (log R r ) 1-Q . Therefore Mod Q (A, B
) is small when diam(A) or diam(B) is small compared to dist(A, B). The Loewner spaces are the spaces for which a kind of a converse inequality occurs:

Definition 4.5. Denote by ∆(A, B) the relative distance between two disjoint compact connected subsets A, B ⊂ Z i.e. ∆(A, B) = dist(A, B) min{diam A, diam B} .
Then Z is a Loewner space if there exits a homeomorphism ϕ : [0, +∞) → [0, +∞) such that for every pair of disjoint compact connected subsets A, B ⊂ Z one has

Mod Q (A, B) ≥ ϕ(∆(A, B) -1 ).
Example 4.6. Basic examples of Loewner spaces include the Euclidean spaces of dimension at least 2, and the Carnot groups (Definition 3.6) equipped with Carnot-Carathéodory metrics (see e.g. [START_REF] Hajlasz | Sobolev met Poincaré[END_REF]). As a consequence, the boundaries of rank-one symmetric spaces different from H 2 R are Loewner, see Proposition 3.7. Among the currently known examples of Loewner spaces, the only ones which arise as boundaries of hyperbolic groups are the boundaries of rank-one symmetric spaces (different from H 2 R ), and the boundaries of Fuchsian buildings [START_REF] Bourdon | Poincaré inequalities and quasiconformal structure on the boundary of some hyperbolic buildings[END_REF]. Just from the definitions, one sees that bi-Lipschitz homeomorphisms preserve the Loewner property. The situation for quasi-Möbius homeomorphisms is more subtle. The following theorem is a combination of results from [START_REF] Tyson | Quasiconformality and quasisymmetry in metric measure spaces[END_REF] and [START_REF] Heinonen | Quasiconformal maps in metric spaces with controlled geometry[END_REF].

Theorem 4.7. Let f : (Z 1 , d 1 , H 1 ) → (Z 2 , d 2 , H 2 ) be a quasi-Möbius homeomor- phism between Alhfors-regular metric spaces of dimensions Q 1 and Q 2 respectively. Suppose Z 1 is Loewner. Then Q 2 ≥ Q 1 . Moreover if Q 2 = Q 1 , then 1. Z 2 is Loewner.
2. f is absolutely continuous with respect to H 1 and H 2 .

M. Bourdon

As an illustration, we mention Corollary 4.8. Suppose S 1 and S 2 are quasi-isometric rank-one non-compact symmetric spaces. Then S 1 = S 2 .

Proof. By Theorem 2.12, the boundaries of S 1 and S 2 are homeomorphic. Thus the boundaries have the same topological dimension. Among rank-one symmetric spaces, H 2 R is the only one whose boundary is a circle. Suppose now that S 1 and S 2 are different from H 2 R . Then their boundaries are Loewner spaces. Thus, Theorem 4.7 implies that the Hausdorff dimensions of ∂S 1 and ∂S 2 are equal. Since the topological and Hausdorff dimensions of ∂H n K are respectively kn-1 and kn+k -2 (Proposition 3.7), the statement follows.

Quasi-conformal homeomorphisms

We now discuss quasi-Möbius homeomorphisms in relation with the classical notion of quasi-conformal homeomorphisms. Let f : Z 1 → Z 2 be a map between metric spaces. For z ∈ Z 1 and 0 < r < diam(Z 1 ) define

L f (z, r) = sup{d(f (z), f (w)) | w ∈ B(z, r)}, l f (z, r) = inf{d(f (z), f (w)) | w ∈ Z 1 \ B(z, r)}. Definition 4.9. A homeomorphism f : Z 1 → Z 2 between metric spaces, is quasi-conformal if there exists a H ≥ 1, such that for every z ∈ Z 1 one has lim sup r→0 L f (z,r) l f (z,r) ≤ H.
Examples of quasi-conformal homeomorphisms include bi-Lipschitz homeomorphisms. It is easy to see that quasi-Möbius homeomorphisms are quasi-conformal as well (with H = η(1)). For the compact Loewner spaces the notions of quasiconformal and quasi-Möbius homeomorphisms are equivalent: Theorem 4.10 (Heinonen-Koskela [START_REF] Heinonen | Quasiconformal maps in metric spaces with controlled geometry[END_REF]). Let Z 1 and Z 2 be compact Loewner spaces with the same Hausdorff dimension Q, and let f : Z 1 → Z 2 be a homeomorphism. The following properties are equivalent.

1. f is H-quasi-conformal. 2. There is a constant C ≥ 1 such that for every family F of curves in Z 1 , one has C -1 Mod Q (F) ≤ Mod Q (f (F)) ≤ C Mod Q (F). 3. f is η-quasi-Möbius.
Moreover the constants H, C and the distortion function η are quantitatively related just in terms of the geometric data of the spaces.

Here is an application to group actions, that will serve in Section 7.

Corollary 4.11 ([33]

). Suppose Z is a compact Loewner space of dimension Q and let H be its Q-Hausdorff measure. Let G be a group acting on Z by uniform quasi-conformal homeomorphisms (i.e. there is

H ≥ 1 such that every g ∈ G is H-quasi-conformal). Suppose that G contains a subgroup Γ such that 1. Γ ⊂ Moeb(Z).
2. The diagonal action of Γ on (Z 2 , H × H) is ergodic.

3. The diagonal action of Γ on

T (Z) := {(z 1 , z 2 , z 3 ) ∈ Z 3 | z i = z j for i = j}
is properly discontinuous and cocompact.

Then G ⊂ Moeb(Z).

Proof of the Corollary. Let G be the closure of G in Homeo(Z). According to Theorem 4.10, G acts on Z by uniform quasi-Möbius homeomorphisms (i.e. we can assume the distortion function to be the same for every g ∈ G). By continuity G also acts by uniform quasi-Möbius homeomorphisms. By Proposition 4.1 it acts properly on T (Z).

We will deduce the corollary from the Sullivan characterisation of Möbius homeomorphisms (Proposition 4.3). Since G acts properly on T (Z), our third assumption implies that Γ is a cocompact lattice in G. In particular the Haar measure on G is bi-invariant. Let m be the finite measure on Γ\G obtained by restricting the Haar measure to a fundamental domain of Γ in G. Let µ be the measure on Z 2 defined in Proposition 4.3. Let s : Γ\G → G be a measurable section. We define a new measure ν on Z 2 by

ν(B) = Γ\G µ(s(g) • B) dm(g)
for every Borel set B ⊂ Z 2 . By the first assumption and Proposition 4.3, the measure µ is Γ-invariant. Since m is invariant under the right action of G, we get that ν is G-invariant. Moreover ν is absolutely continuous with respect to µ. Indeed, the group G acts on Z by quasi-Möbius homeomorphisms, and quasi-Möbius homeomorphisms are absolutely continuous with respect to H (Proposition 4.7). With our second assumption we obtain that µ is equal to Cν for some C > 0. Therefore µ is G-invariant and we conclude by using Sullivan's characterization (Proposition 4.3).

Mostow rigidity

This section is devoted to the proof of Mostow's theorem. We also present two generalizations: the Besson-Courtois-Gallot theorem (Theorem 5.2), and a generalization of Mostow's theorem to quasi-convex geometric actions (Theorem 5.4). The section ends with a survey of several related results.

Mostow theorem

We state Mostow's theorem (for rank-one non-compact symmetric spaces), and we explain how it can be deduced from results of the previous sections.

As usual we normalize the Riemannian metric of the rank-one symmetric spaces so that the maximum of the sectional curvatures is -1.

Theorem 5.1. Let S 1 and S 2 be rank-one symmetric spaces different from H 2 R . For i = 1, 2 let Γ i be a lattice in Isom(S i ). Then any group isomorphism ϕ : Γ 1 → Γ 2 is the conjugacy by an isometry from S 1 to S 2 .

Mostow proved the cocompact lattice case [START_REF] Mostow | Strong rigidity of locally symmetric spaces[END_REF]. G. Prasad extended the proof to finite covolume lattices [START_REF] Prasad | Strong rigidity of Q-rank 1 lattices[END_REF].

Proof. Suppose first that Γ 1 and Γ 2 are cocompact lattices. Then Γ i acts geometrically on S i . By Svarc-Milnor (Proposition 2.3), the isomorphism ϕ induces a ϕ-equivariant quasi-isometry from S 1 to S 2 . It extends to a ϕ-equivariant quasi-Möbius homeomorphism f : ∂S 1 → ∂S 2 (see Theorem 2.12). Since S 1 and S 2 are rank-one symmetric spaces different from H 2 R , their boundaries (equipped with the visual metrics d CAT ) are Loewner spaces. Thus from Theorem 4.7, the Hausdorff dimensions of ∂S i are equal and f is absolutely continuous with respect to the Hausdorff measures. Therefore f × f is absolutely continuous with respect to the measures µ i defined in Proposition 4.3. Since µ i is Γ i -invariant and ergodic (Theorem 2.14), and since f × f is equivariant, we get that (f × f ) * µ 2 = Cµ 1 for some constant C > 0. Therefore Sullivan's characterization (Proposition 4.3) implies that f is a Möbius homeomorphism. Finally f is the boundary extension of an isometry by Theorem 2.19.

In the non-cocompact case, Prasad [START_REF] Prasad | Strong rigidity of Q-rank 1 lattices[END_REF] proved that the equivariant quasiisometry from S 1 to S 2 still exists, by using informations about the cusps. The rest of the proof is similar. This proof, arising from the Sullivan's ergodic approach [START_REF] Sullivan | Discrete conformal groups and measurable dynamics[END_REF], is similar in spirit to Mostow's original proof [START_REF] Mostow | Strong rigidity of locally symmetric spaces[END_REF]. At the end, Mostow's argument is different, it relies -in a delicate way -on absolute continuity and ergodicity, to show that f preserves the R-circles (i.e. boundaries of curvature -1 totally geodesic planes). Then he deduces from this property that f is the boundary extension of an isometry.

Besson-Courtois-Gallot theorem

A remarkable generalization of Mostow's theorem is due to Besson, Courtois, Gallot [START_REF] Besson | Entropies et rigidités des espaces localement symétriques de courbure strictement négative[END_REF][START_REF] Besson | Minimal entropy and Mostow's rigidity theorems[END_REF].

Let M be a compact connected Riemannian manifold M and let M be its universal cover. Pick O ∈ M and define the volume entropy of M (independent of O) by

h(M ) := lim R→+∞ 1 R ln Vol(B(O, R)),
where Vol(B(O, R)) denotes the volume in M of the ball B(O, R).

Theorem 5.2. Let M 0 and M be compact connected Riemannian manifolds of the same dimension n. Suppose M 0 is a locally symmetric manifold of negative sectional curvature. Then for every non-zero degree continuous map f : M → M 0 , one has the following inequality

h(M ) n Vol(M ) ≥ | deg(f )|h(M 0 ) n Vol(M 0 ).
Moreover, for n ≥ 3, the equality holds if and only if M is a locally symmetric space and f is homotopic to a homothetic covering.

The theorem admits several important applications in geometry, topology and dynamics. We refer to [START_REF] Besson | Entropies et rigidités des espaces localement symétriques de courbure strictement négative[END_REF][START_REF] Besson | Minimal entropy and Mostow's rigidity theorems[END_REF] for more informations (see also subsection 5.4). It has been generalized to finite volume manifolds by Boland, Connell, Souto [START_REF] Boland | Volume rigidity for finite volume manifolds[END_REF].

Quasi-convex geometric actions

We present a generalization of Mostow's theorem to quasi-convex geometric actions on CAT(-1)-spaces.

Definition 5.3. Let X be a proper hyperbolic space. A subset Y ⊂ X is quasiconvex if there is a constant R ≥ 0 such that for every pair of points y 1 , y 2 ∈ Y the geodesic segments [y 1 , y 2 ] ⊂ X lie in N R (Y ). A finitely generated group Γ acts on X quasi-convex geometrically if it acts by isometries, properly discontinuously, and if its orbits are quasi-convex subsets of X. The limit set of Γ is the following subset of ∂X (independent of O ∈ X)

Λ = Γ • O X∪∂X ∩ ∂X.
Several properties of geometric actions generalize to quasi-convex geometric actions. In particular such groups are hyperbolic. The orbit map g ∈ Γ → g •O ∈ X is a quasi-isometric embedding. It extends canonically to a quasi-Möbius homeomorphism from ∂Γ to Λ.

The following result was first proved by U. Hamenstädt [START_REF] Hamenstädt | Entropy-rigidity of locally symmetric spaces of negative curvature[END_REF] for geometric actions on simply connected Riemannian manifold of curvature at most equal to -1.

Theorem 5.4. Let S be a rank-one symmetric space different from H 2 R . Suppose its Riemannian metric is normalized so that the maximum of the sectional curvatures is -1. Let Γ be a cocompact lattice in Isom(S). Assume that Γ acts quasi-convex geometrically on a CAT(-1)-space X and let Λ ⊂ ∂X be its limit set. Then Hausdim(∂S) ≤ Hausdim(Λ). Moreover the equality holds if and only if there exists a Γ-equivariant totally geodesic isometric embedding of S into X.

The inequality is due to Pansu [START_REF] Pansu | Dimension conforme et sphère à l'infini des variétés à courbure négative[END_REF]. It is also a consequence of Theorem 4.7 since ∂S and Λ are quasi-Möbius homeomorphic. The equality case is proved in [START_REF] Bourdon | Sur le birapport au bord des CAT(-1)-espaces[END_REF] (the case X is a rank-one symmetric space has been treated independently by C. Yue [START_REF] Yue | Dimension and reigidity of quasi-Fuchsian representation[END_REF]). The proof is similar to the one we gave for the Mostow theorem, it relies on Theorem 4.7. Theorem 5.4 has been generalized in [START_REF] Xie | A Bowen type rigidity theorem for non-cocompact hyperbolic groups[END_REF][START_REF] Das | Tukia's isomorphism theorem in CAT(-1)-spaces[END_REF] to geometrically finite actions of finite covolume lattices of Isom(S).

When S = H 2 R and Γ is a lattice in Isom(H 2 R ), the same statement holds apart from the fact that the isometric embedding is not Γ-equivariant in general. This result is due to R. Bowen [START_REF] Bowen | Hausdorff dimension of quasi-circles[END_REF] when X = H 3 R , and to M. Bonk and B. Kleiner [START_REF] Bonk | Rigidity for quasi-Fuchsian actions on negatively curved spaces[END_REF] for general CAT(-1)-spaces X.

Further results

1) One expects that some versions of Mostow's rigidity hold for quite general hyperbolic spaces. For instance, given "reasonable" hyperbolic spaces X 1 and X 2 , and a group Γ acting on them geometrically, suppose that the canonical boundary map is absolutely continuous with respect to the Hausdorff measures (recall that the Hausdorff measure class is independent of the choice of a visual metric). Does this imply that there exists a Γ-equivariant homothety from X 1 to X 2 ?

When X 1 = X 2 = H 2 R , this was established by T. Kuusalo [START_REF] Kuusalo | Boundary mappings of geometric isomorphisms of Fuchsian groups[END_REF]. When X 1 and X 2 are metric trees, this is a result of Coornaert [START_REF] Coornaert | Rigidité ergodique de groupes d'isométries d'arbres[END_REF]. C. Croke [START_REF] Croke | Rigidity of surfaces of nonpositive curvature[END_REF] and J-P. Otal [START_REF] Otal | Le spectre marqué des longueurs des surfaces à courbure négative[END_REF] proved the case X 1 and X 2 are simply connected Riemannian surfaces of negative curvature. Hersonsky-Paulin [START_REF] Hersonsky | On the rigidity of dicrete isometry groups of negatively curved spaces[END_REF] generalized it to negatively curved Riemannian surfaces with singularities. When X 1 is a rank-one symmetric space, and X 2 is a simply connected negatively curved Riemannian manifold, this is due to Hamenstädt [START_REF] Hamenstädt | Cocycles, symplectic structures and intersections[END_REF] (by using the Besson-Courtois-Gallot theorem).

We remark that the above problem admits several equivalent formulations (in terms of cross-ratio, marked length spectrum, geodesic flow, etc), see [START_REF] Ledrappier | Structure au bord des variétés à courbure négative[END_REF][START_REF] Hamenstädt | Cocycles, Hausdorff measures and cross ratios[END_REF].

In [START_REF] Mineyev | Metric conformal structures and hyperbolic dimension[END_REF], I. Mineyev constructed Isom(X)-invariant conformal structures and cross-ratios on boundaries of hyperbolic spaces X, that generalize the CAT(-1) setting.

K. Biswas [START_REF] Biswas | On Moebius and conformal maps between boundaries of CAT(-1)-spaces[END_REF] proved that every Möbius homeomorphism between the boundaries of proper geodesically complete CAT(-1)-spaces X 1 and X 2 extends to a (1, log 2)-quasi-isometry from X 1 to X 2 with 1 2 log 2-dense image in X 2 . 2) P. Storm [START_REF] Storm | The barycenter method on singular spaces[END_REF] proved a version of the Besson-Courtois-Gallot theorem for manifolds with boundary. As a consequence he solved the following conjecture of Bonahon. Let ρ 0 , ρ 1 be quasi-convex geometric actions of a group Γ on H n R with n ≥ 3. Consider, for i = 0, 1, the convex hull H i ⊂ H n R of the limit set Λ(ρ i ) ⊂ S n-1 , and suppose that the boundary of H 0 in H n R is totally geodesic. Then Vol(H 0 /ρ 0 (Γ)) ≤ Vol(H 1 /ρ 1 (Γ)). Moreover the equality holds if and only if ρ 0 and ρ 1 are conjugate in Isom(H n R ). We note that for such ρ 0 , ρ 1 , it is commonly conjectured that Hausdim(Λ(ρ 0 )) ≤ Hausdim(Λ(ρ 1 )), and that the equality holds if and only if ρ 0 and ρ 1 are conjugate in Isom(H n R ).

3) Divergence groups form a vast generalization of quasi-convex geometric actions on hyperbolic spaces

X. A subgroup Γ ⊂ Isom(X) is a divergence group if it acts properly discontinuously on X, if the critical exponent δ(Γ) := inf{s > 0 | g∈Γ exp(-sd(O, g • O)) < +∞}
is finite, and if the sum g∈Γ exp(-sd(O, g • O)) diverges at s = δ(Γ). See [START_REF] Hersonsky | On the rigidity of dicrete isometry groups of negatively curved spaces[END_REF] for several examples of such groups.

The limit set of a divergence group carries a natural finite measure called the Patterson-Sullivan measure (it coincides with the Hausdorff measure when Γ acts quasi-convex geometrically). Its ergodic properties are well studied (see for instance [START_REF] Sullivan | Discrete conformal groups and measurable dynamics[END_REF], [START_REF] Sullivan | Entropy, Hausdorff measures old and new, and limit sets of geometrically finite Kleinian groups[END_REF], [START_REF] Burger | CAT(-1)-spaces, divergence groups and their commensurators[END_REF], [START_REF] Roblin | Ergodicité et équidistribution en courbure négative[END_REF]).

Sullivan [START_REF] Sullivan | Discrete conformal groups and measurable dynamics[END_REF], Burger-Mozes [START_REF] Burger | CAT(-1)-spaces, divergence groups and their commensurators[END_REF] and Yue [START_REF] Yue | Mostow rigidity of rank 1 discrete groups with ergodic Bowen-Margulis measure[END_REF] obtained several rigidity theorems for such actions. In particular, the following result is established in [START_REF] Burger | CAT(-1)-spaces, divergence groups and their commensurators[END_REF]. Let X 1 and X 2 be proper CAT(-1)-spaces and let Γ ∈ Isom(X 1 ) be a divergence group. Let Λ ⊂ ∂X 1 be its limit set. Then, for every group homomorphism ρ : Γ → Isom(X 2 ) with non-elementary image, there is a unique ρ-equivariant measurable map f : Λ → ∂X 2 , and almost all values of f belong to the limit set of ρ(Γ).

Hersonsky-Paulin [START_REF] Hersonsky | On the rigidity of dicrete isometry groups of negatively curved spaces[END_REF] obtained Mostow type theorems for divergence groups of CAT(-1)-spaces, under the assumption that the above map is absolutely continuous with respect to the Patterson-Sullivan measures.

4)

In [START_REF] Sullivan | Discrete conformal groups and measurable dynamics[END_REF], Sullivan addressed the following problem. Suppose Γ is a finitely generated subgroup of Isom(H n R ) such that the topological dimension of its limit set Λ is equal to its Hausdorff dimension. Does it imply that Λ is a round sphere?

For quasi-convex subgroups, this was established by Yue [START_REF] Yue | Dimension and reigidity of quasi-Fuchsian representation[END_REF] (in the statement he assumed that Λ is a topological sphere, but his argument does not need the latter assumption). M. Kapovich [START_REF] Kapovich | Homological dimension and critical exponent of Kleinian groups[END_REF] proved it for geometrically finite groups, and T. Das, D. Simmons and M. Urbańsky [START_REF] Das | Dimension rigidity in conformal structures[END_REF] for even more general Kleinian groups.

Bonk and Kleiner obtained the following generalization [START_REF] Bonk | Rigidity for quasi-Möbius group actions[END_REF][START_REF] Bonk | Rigidity for quasi-Fuchsian actions on negatively curved spaces[END_REF]. Suppose that a group Γ acts quasi-convex geometrically on a CAT(-1)-space X, and let Λ ⊂ ∂X be its limit set. Let n ≥ 1 be the topological dimension of Λ. Then Hausdim(Λ) ≥ n, and equality holds if and only if Γ acts geometrically on an isometric copy of H n+1 R in X. K. Kinneberg [START_REF] Kinneberg | Rigidity for quasi-Möbius actions on fractal metric spaces[END_REF] established a coarse version of this result. Suppose a group Γ acts geometrically on an AC u (-1)-metric space X. These are the metric spaces with asymptotic upper curvature -1, a geometric property that is invariant by rough isometries, and that has been introduced by Bonk and Foertsch [START_REF] Bonk | Asymptotic upper curvature bounds in coarse geometry[END_REF]. If ∂X is homeomorphic to S n with n ≥ 2, and if the volume entropy of X is at most equal to n, then Γ is virtually a cocompact lattice in Isom(H n+1 R ) and X is roughly isometric to H n+1 R . Kapovich considered the following situation. Let Γ be a discrete virtually torsion free subgroup of Isom(H n R ) and let δ(Γ) be its critical exponent as defined above (when Γ is geometrically finite, δ(Γ) is equal to the Hausdorff dimension of the limit set [START_REF] Sullivan | Entropy, Hausdorff measures old and new, and limit sets of geometrically finite Kleinian groups[END_REF]). Kapovich proved in [START_REF] Kapovich | Homological dimension and critical exponent of Kleinian groups[END_REF] that δ(Γ) + 1 is at least equal to the virtual homological dimension of the pair (Γ, Π), where Π denotes a set of representatives of the conjugacy classes of maximal virtually abelian subgroups of virtual rank at least two. Its proof uses Besson-Courtois-Gallot techniques. When Γ ⊂ Isom(H n R ) is discrete finitely generated, he conjectures that δ(Γ) + 1 is at least equal to the virtual cohomological dimension of the pair (Γ, Π), and that equality holds if and only if Γ is geometrically finite and its limit set is a round sphere of dimension δ(Γ).

The zoom method

The convergence property (Proposition 4.1) allows one to "zoom in" at some point of the space by using a sequence of uniformly quasi-Möbius homeomorphisms. Inspired by Mostow's theorem, J. Ferrand [START_REF] Ferrand | Transformations conformes et quasi-conformes des variétés riemanniennes compactes[END_REF][START_REF] Ferrand | Geometrical interpretations of scalar curvature and regularity of conformal homeomorphisms[END_REF] exploited this idea to solve the Lichnérowicz conjecture, which states that the conformal group of a compact manifold M is compact, unless M is conformally equivalent to an Euclidean sphere. Later, Tukia [START_REF] Tukia | On quasiconformal groups[END_REF] used the zoom method to prove that every finitely generated group quasi-isometric to H n R (with n ≥ 3) is virtually isomorphic to a cocompact lattice in Isom(H n R ). This section is devoted to these results and their proofs. We also present a rather elementary proof of Mostow's theorem in the real case, based on the zoom method, and due to Tukia [START_REF] Tukia | Differentiability and rigidity of Möbius groups[END_REF]. The section ends with a survey of several related results.

Ferrand theorem

The following theorem solves the generalized Lichnérowicz conjecture. Theorem 6.1. Let M be a Riemannian manifold of dimension n ≥ 2. Denote by Conf(M ) the group of conformal diffeomorphisms of M . Then Conf(M ) acts properly on M unless M is conformally equivalent to the Euclidean n-sphere or n-space.

The compact manifolds case was treated by Ferrand in 1969-71 in [START_REF] Ferrand | Transformations conformes et quasi-conformes des variétés riemanniennes compactes[END_REF][START_REF] Ferrand | Geometrical interpretations of scalar curvature and regularity of conformal homeomorphisms[END_REF] (Obata [START_REF] Obata | The conjectures on conformal transformations of Riemannian manifolds[END_REF] proved it too, but only for Conf 0 (M )). Soon after, D. V. Alekseevskii proposed a proof for the non-compact manifolds. His proof was accepted for more than twenty years until R. Zimmer and K. Gutshera found a serious gap. Finally Ferrand solved the non-compact case in [START_REF] Ferrand | The action of conformal transformations on a Riemannian manifold[END_REF] by introducing new global conformal invariants. See [START_REF] Ferrand | Histoire de la réductibilité du groupe conforme des variétés riemanniennes (1964-1994[END_REF] for a detailed story of the Lichnérowicz conjecture.

We will only prove Theorem 6.1 for compact manifolds, since in this case all the geometric tools have been already defined. For an alternative proof, which makes use of the Weyl tensor, see [START_REF] Frances | Autour du théorème de Ferrand-Obata[END_REF]. The generalized Lichnérowicz conjecture is discussed in [START_REF] Haissinsky | Quasi-Moebius group actions[END_REF].

Proof of the compact manifold case. A compact Riemannian manifold is a Loewner space, and a conformal diffeomorphism is a 1-quasi-conformal homeomorphism. Therefore Theorem 4.10 implies that Conf(M ) acts by uniform quasi-Möbius homeomorphisms (i.e. we can assume the distortion function η to be the same for every g ∈ Conf(M )).

Thus the convergence property (Proposition 4.1) in combination with the fact that Conf(M ) is closed in Homeo(M ) (see [START_REF] Ferrand | Geometrical interpretations of scalar curvature and regularity of conformal homeomorphisms[END_REF][START_REF] Capogna | Conformality and Q-harmonicity in sub-Riemannian manifolds[END_REF]), implies the following characterization : Conf(M ) is not compact if and only if there exists a sequence (g k ) k∈N of elements in Conf(M ), and points a, b ∈ M (possibly a = b), such that for every compact subset K ⊂ M \ {a} the sequence (g k ) k∈N converges uniformly on K to the constant map b.

Assume now that Conf(M ) is not compact and consider (g k ) k∈N , a, b as above. Pick three pairwise distinct points a 1 , a 2 , a 3 ∈ K \ {a}. We claim that for every permutation α, β, γ of 1, 2, 3 and k large enough:

d(g k (a α ), g k (a β )) d(g k (a α ), g k (a γ )). Indeed, let c ∈ M \ {b} and c k = g -1 k (c). One has c k → a when k → +∞. Since g k (a i ) → b for i = 1, 2, 3, we have for k large enough, d(g k (a α ), g k (a β )) d(g k (a α ), g k (a γ )) d(g k (a α ), g k (a β ))d(c, g k (a γ )) d(g k (a α ), g k (a γ ))d(c, g k (a β )) ≤ η d(a α , a β )d(c k , a γ ) d(a α , a γ )d(c k , a β ) 1.
Let exp : T b M → M be the Riemann exponential map. For every compact subset K ⊂ M \ {a} the map exp -1 •g k is well defined on K for k large enough. The previous discussion shows that for every permutation α, β, γ of 1, 2, 3 and k large enough

exp -1 •g k (a α ) -exp -1 •g k (a β ) exp -1 •g k (a α ) -exp -1 •g k (a γ ) .
Thus there exist λ k → +∞ when k → +∞, and affine homotheties h k of T b M , with ratios λ k , so that for k large enough we have h k • exp -1 •g k (a 1 ) = 0, and

h k • exp -1 •g k (a i ) -h k • exp -1 •g k (a j ) 1,
for every distinct i, j ∈ {1, 2, 3}. The sequence (h k • exp -1 •g k ) is uniformly quasi-Möbius on K and normalized on the three points a 1 , a 2 , a 3 . Therefore, by the convergence property (Lemma 4.2), up to a subsequence, it converges uniformly on K to a quasi-Möbius map f K : K → T b M , where T b M denotes the one point compactification of T b M . Moreover, since (g k ) k∈N converges uniformly on K to the constant map b, and since the tangent map of exp -1 at b is the identity, f K is a 1-quasi-conformal homeomorphism onto its image. By considering an exhaustion (K i ) i∈N of M \ {a}, a diagonal argument shows that there is a quasi-Möbius map f : M \ {a} → T b M , which is a 1-quasi-conformal homeomorphism onto its image. We now establish that f (M \ {a}) is equal to the sphere T b M minus a point. The subset

U := f (M \ {a}) is open in T b M . Suppose by contradiction that U is different from T b M minus a point. Then U \ U contains at least two distinct points z, w. Let z k , w k ∈ U with z k → z and w k → w when k → +∞. The points f -1 (z k ) and f -1 (w k ) tend to a; thus [a 1 , f -1 (z k ), a 2 , f -1 (w k )] tends to 0. But [f (a 1 ), z k , f (a 2 ), w k ] tends to [f (a 1 ), z, f (a 2 ), w] = 0, which contradicts the fact that f is a quasi-Möbius map.
Therefore f extends to a quasi-Möbius homeomorphism from M to the Euclidean n-sphere, which is 1-quasi-conformal on M \{a}. Such a map is a conformal diffeomorphism [START_REF] Ferrand | Geometrical interpretations of scalar curvature and regularity of conformal homeomorphisms[END_REF][START_REF] Capogna | Conformality and Q-harmonicity in sub-Riemannian manifolds[END_REF]. The proof is complete.

Sullivan-Tukia theorem

The following result characterizes the groups which are quasi-isometric to H n R .

Theorem 6.2 (Sullivan [START_REF] Sullivan | On the ergodic theory at infinity of an arbitrary discrete group of hyperbolic motions[END_REF] for n = 3, Tukia [START_REF] Tukia | On quasiconformal groups[END_REF] in general). Let Γ be a finitely generated group quasi-isometric to H n R , with n ≥ 3. Then there exists a cocompact lattice Φ ⊂ Isom(H n R ) and a surjective homomorphism of groups Γ → Φ with finite kernel.

The above statement holds for every rank-one non-compact symmetric space too. Its proof for the hyperbolic plane H 2 R relies on works by Tukia [START_REF] Tukia | Homeomorphic conjugates of Fuchsian groups[END_REF], Gabai [START_REF] Gabai | Convergence groups are Fuchsian groups[END_REF], Casson and Jungreis [START_REF] Casson | Convergence groups and Seifert fibered 3manifolds[END_REF]. An alternative argument, using Perelman's proof of Thurston's geometrization conjecture, is given in [START_REF] Drutu | Lectures on Geometric Group Theory[END_REF]. In [START_REF] Markovic | Criterion for Cannon's conjecture[END_REF], V. Markovic gives a proof based on the recent work of Agol-Wise on cube complexes. The result for the hyperbolic complex spaces H m C has been established by Chow [START_REF] Chow | Groups quasi-isometric to complex hyperbolic space[END_REF]. For the remaining rank-one symmetric spaces, it follows from Pansu's theorem [START_REF] Pansu | Métriques de Carnot-Carathéodory et quasiisométries des espaces symétriques de rang un[END_REF], see Theorem 7.1.

We remark that in addition to Tukia's original paper, there are several other expositions of Tukia's proof in the literature, e.g. [START_REF] Bourdon | Quasi-conformal geometry and Mostow rigidity[END_REF], [START_REF] Kapovich | Lectures on Quasi-Isometric Rigidity[END_REF], [START_REF] Drutu | Lectures on Geometric Group Theory[END_REF].

Consider the isometric action of Γ on itself by left translations. Since Γ is quasi-isometric to H n R , every element of Γ induces a quasi-isometry of H n R , which is unique up to bounded distance, and with uniform quasi-isometry constants. Thus by Theorems 2.12 and 4.10, Γ acts on S n-1 = ∂H n R by uniform quasi-conformal homeomorphisms. The kernel of this action is finite. We still denote by Γ the quotient by the kernel.

We first search for a Γ-invariant structure on S n-1 . A measurable field of ellipsoids on S n-1 is a measurable map which assigns to a.e. z ∈ S n-1 an (n -2)ellipsoid centered at 0 in T z S n-1 .

We are only concerned with non-degenerate ellipsoids, up to homothety, and centered at 0. The space of such ellipsoids in R n-1 is the symmetric space X := SL n-1 (R)/SO(n -1).

Since quasi-conformal homeomorphisms of S n-1 are differentiable a.e. (Rademacher-Stepanov's Theorem [START_REF]Lectures on n-dimensional quasiconformal mappings[END_REF]) and absolutely continuous (Theorems 4.10 and 4.7), every quasi-conformal homeomorphism f : S n-1 → S n-1 acts on the space of measurable fields of ellipsoids, as follows. If ξ = {ξ z } z∈S n-1 is a measurable field of ellipsoids, then :

(f * ξ) z := D f -1 (z) f (ξ f -1 (z) ).
Lemma 6.3. There exists a bounded Γ-invariant measurable field of ellipsoids on S n-1 .

Proof of the lemma. For a.e. z ∈ S n-1 , let

E z = D g -1 (z) g(S g -1 z ) | g ∈ Γ ,
where S z is the unit sphere in T z S n-1 . By choosing a measurable trivialisation of the orthonormal frame bundle of S n-1 , each set E z can be identified with a subset of the symmetric space X defined above. In addition we have for g ∈ Γ, and a.e.

z ∈ S n-1 E g(z) = D z g(E z ),
where D z g acts on X by isometry (indeed SL n-1 (R) does). The eccentricities of the ellipsoids in E z are bounded by the uniform quasi-conformal constant of the Γ-elements. Thus E z is a bounded subset of X. Every bounded subset A in a Hadamard manifold, admits a well-defined canonical "center", namely the center of the unique smallest closed ball containing A. Define ξ z to be the center of E z . The field {ξ z } z∈Z possesses the expected properties.

Let ξ = {ξ z } z∈S n-1 be a bounded Γ-invariant measurable field of ellipsoids. Our goal is now to construct a quasi-conformal homeomorphism f of S n-1 such that f Γf -1 ⊂ Conf(S n-1 ).

For n = 3, Sullivan observed that the measurable Riemann mapping theorem implies that there exists a quasi-conformal homeomorphism f of S 2 such that f * ξ is a field of circles. Therefore f Γf -1 is a group of quasi-conformal homeomorphisms that are 1-quasi-conformal a.e. Such homeomorphisms are conformal diffeomorphisms (see e.g. [START_REF] Tukia | A remark on 1-quasiconformal maps[END_REF]). Thus f admits the expected property.

When n ≥ 4, the measurable Riemann mapping theorem is not valid. Instead, Tukia proposed the following argument based on the zoom method.

The field ξ is measurable, so it is approximately continuous a.e. In other words for a.e. z ∈ S n-1 and every ε > 0, we have

lim r→0 H w ∈ B(z, r) | d X (ξ z , ξ w ) < ε /H(B(z, r)) = 1,
where H denotes the spherical measure on S n-1 . Let z 0 ∈ S n-1 such that ξ is approximately continuous at z 0 . By conjugating Γ by a projective isomorphism of S n-1 if necessary, we may assume that ξ z0 is a round sphere. Let O ∈ H n R be an origin, and let (g k ) k∈N be a sequence in Γ so that

g k • O → z 0 when k → ∞, and dist H n R (g k • O, [O, z 0 )
) is uniformly bounded. (The existence of (g k ) k∈N comes from the fact that Γ and H n R are quasi-isometric.) Let h k ∈ Isom(H n R ) be a loxodromic element whose axis contains the ray [O, z 0 ) and such that (

h k • g k ) • O is uniformly close to O. By considering an ideal triangle ∆(a, b, c) ⊂ H n R centered at O,

and its images by the h

k • g k 's, it follows from Lemma 4.2 that (h k • g k ) k∈N subconverges to a quasi-conformal homeomorphism f . Since ξ is Γ-invariant, the group (h k • g k )Γ(h k • g k ) -1 leaves invariant the field h k * ξ.
Because ξ is approximately continuous at z 0 and ξ z0 is a round sphere, the sequence (h k * ξ) k∈N converges in measure to the field of round spheres. Therefore for every g ∈ Γ, the eccentricity of the differential of (

h k • g k ) • g • (h k • g k ) -1
converges in measure to the constant function 1. This implies that the limit map f • g • f -1 is a conformal diffeomorphism (see [START_REF] Tukia | On quasiconformal groups[END_REF] Lemma B2 for more details). Thus we have f Γf -1 ⊂ Conf(S n-1 ).

We know that Conf(S n-1 ) = Moeb(S n-1 ) = Isom(H n R ). Indeed the first equality follows from Liouville's theorem. The second one is a standard result in classical hyperbolic geoemetry (it is also a particular case of Theorem 2.19). Thus we obtain that f Γf -1 ⊂ Isom(H n R ). It remains to prove that f Γf -1 is a cocompact lattice of Isom H n R . By Proposition 2.15, Γ acts properly discontinuously and cocompactly on the space of triples Isom(S 2 ) respectively. Then, for any quasi-isometry F : Γ 1 → Γ 2 , there exists a homothety h : S 1 → S 2 and finite-index subgroups Φ i ⊂ Γ i (i = 1, 2), such that h conjugates Φ 1 to Φ 2 and lies within finite distance from F . This result is in keeping with the following general problem called pattern rigidity. For i = 1, 2, let Γ i be a finitely generated group and let H i be a finite collection of quasi-convex subgroups of Γ i . Suppose we are given a quasi-isometry F : Γ 1 → Γ 2 and a constant R ≥ 0 that enjoy the following properties :

1. For every g 1 ∈ Γ 1 and every H 1 ∈ H 1 there exists g 2 ∈ Γ 2 and H 2 ∈ H 2 such that F (g 1 H 1 ) and g 2 H 2 lie within Hausdorff distance at most R.

2. For every g 2 ∈ Γ 2 and every H 2 ∈ H 2 there exists g 1 ∈ Γ 1 and H 1 ∈ H 1 such that F (g 1 H 1 ) and g 2 H 2 lie within Hausdorff distance at most R.

Do there exist finite-index subgroups Φ i ⊂ Γ i and an isomophism ϕ : Φ 1 → Φ 2 that lies at bounded distance from F ? The above Schwartz theorem solves this problem in the case the Γ i 's are noncocompact lattices in Isom(H n K ) and the H i 's are the patterns of the parabolic subgroups. The question of pattern rigidity has also been solved by Schwartz [START_REF] Schwartz | Symmetric patterns of geodesics and automorphisms of surface groups[END_REF] for Γ a cocompact lattice in Isom(H n K ) and H a pattern associated to a finite collection of closed geodesics. In [START_REF] Biswas | Pattern rigidity in hyperbolic spaces: duality and PD subgroups[END_REF] Biswas and Mj generalize Schwartz's result to certain duality subgroups of Γ. Biswas [START_REF] Biswas | Flows, fixed points and rigidity for Kleinian groups[END_REF] completely solved the pattern rigidity problem for Γ a cocompact lattice in Isom(H n R ) and H any infinite quasi-convex subgroup of infinite index. Mj [START_REF] Mj | Pattern rigidity and the Hilbert-Smith conjecture[END_REF] proved the following non-linear pattern rigidity result. Let Γ be a hyperbolic Poincaré duality group and let H be a quasi-convex codimension one filling subgroup. Suppose ∂Γ carries a visual metric d such that Hausdim(∂Γ, d) ≤ Topdim(∂Γ) + 2.

Then the pattern-preserving quasi-isometry group of (Γ, H) is a finite extension of Γ.

2) The notion of zooming into the space at a point is exploited by Bonk and Kleiner in a somewhat different way in [START_REF] Bonk | Rigidity for quasi-Möbius group actions[END_REF][START_REF] Bonk | Quasi-hyperbolic planes in hyperbolic groups[END_REF].

Recall that a weak tangent space of a metric space (Z, d) is the Gromov-Hausdorff limit (as k → ∞) of a sequence of pointed metric spaces of the form (Z, z k , 1 ε k d), where z k ∈ Z and ε k → 0. See [START_REF] Burago | A course in metric geometry[END_REF] for more details. Suppose that Γ is a hyperbolic group, and let d be a visual metric on ∂Γ. Bonk and Kleiner deduce from the convergence property (Subsection 4.1) that every weak tangent space of (∂Γ, d) is quasi-Möbius homeomorphic to (∂Γ \ {w}, d), for some w ∈ ∂Γ. In combination with previous works of B. Bowditch and G. Swarup [START_REF] Bowditch | Cut points and canonical splittings of hyperbolic groups[END_REF][START_REF] Swarup | On the cut point conjecture[END_REF], they show that ∂Γ is linearly locally connected as soon as it is connected. In other words there exists C ≥ 1 such that 1. For every ball B(z, r) ⊂ ∂Γ and every pair {w 1 , w 2 } ⊂ B(z, r), there exists an arc γ ⊂ B(z, Cr) that joins w 1 to w 2 .

2. For every ball B(z, r) ⊂ ∂Γ and every {w 1 , w 2 } ⊂ ∂Γ \ B(z, r), there exists an arc γ ⊂ ∂Γ \ B(z, r C ) that joins w 1 to w 2 .

M. Bourdon

Further developments on connectedness properties of ∂Γ can be found in [START_REF] Bonk | Rigidity for quasi-Möbius group actions[END_REF][START_REF] Bonk | Quasi-hyperbolic planes in hyperbolic groups[END_REF][START_REF] Mackay | Spaces and groups with conformal dimension greater than one[END_REF][START_REF] Piaggio | Conformal dimension and canonical splittings of hyperbolic groups[END_REF].

3) The Sullivan-Tukia theorem can also be stated as follows. Suppose that Γ is a hyperbolic group whose boundary is quasi-Möbius homeomorphic to the Euclidean sphere S n-1 , with n ≥ 3. Then Γ is virtually a cocompact lattice in Isom(H n R ). Bonk and Kleiner have considerably weakened the above assumption. They proved in [START_REF] Bonk | Rigidity for quasi-Möbius group actions[END_REF] that if the boundary of a hyperbolic group Γ is quasi-Möbius homeomorphic to an (n -1)-regular metric space of topological dimension n -1, with n ≥ 3, then Γ is virtually a cocompact lattice in Isom(H n R ). 4) P. Haïssinsky [START_REF] Haissinsky | Hyperbolic groups with planar boundaries[END_REF] recently proved the following generalization of Sullivan's theorem: every finitely generated group that admits a quasi-isometric embedding into H 3 R , contains a finite-index subgroup that acts quasi-convex geometrically on H 3 R (see Definition 5.3). 5) T. Dymarz and X. Xie [START_REF] Dymarz | Day's fixed point theorem, Group Cohomology and Quasi-isometric rigidity[END_REF] established Sullivan-Tukia type theorems for actions of amenable groups on the boundary of certain (non symmetric) negatively curved homogeneous manifolds M . In particular, for M = R n A R where A is an expansive matrix diagonalisable over C, they proved that every amenable group Γ that acts on M by uniform quasi-conformal homeomorphisms and cocompactly on ∂ 3 M , is conjugate to a conformal group. They obtained applications to quasiisometry rigidity of lattices in certain solvable Lie groups like cyclic extensions of abelian groups.

Rigidity of quasi-isometries

The proof of the Mostow theorem shows that a quasi-isometry of a rank-one symmetric space (different from H 2 R ) which is equivariant with respect to a lattice, lies within bounded distance from an isometry. What about the non-equivariant quasi-isometries? This section gives examples of hyperbolic spaces where every quasi-isometry lies within bounded distance from an isometry. In particular we will give some ideas of the proof of the following theorem. Theorem 7.1 (Pansu [START_REF] Pansu | Métriques de Carnot-Carathéodory et quasiisométries des espaces symétriques de rang un[END_REF]).

Let S = H n Q with n ≥ 2, or H 2 O .
Then any quasiisometry of S lies within bounded distance from an isometry.

We note that Theorem 7.1 is false for H n R and H n C (see [START_REF] Korányi | Quasiconformal mappings on the Heisenberg group[END_REF] for the complex case). Observe that Theorem 7.1, in combination with Proposition 2.15, implies that the Sullivan-Tukia theorem (Theorem 6.2) holds for H n Q and H 2 O . Subsections 7.1 and 7.2 are devoted to the proof of Theorem 7.1. Subsection 7.3 discusses quasi-isometry rigidity of Fuchsian buildings. Subsection 7.4 contains a survey on several related results.

Differentiability in Carnot groups

The boundary of H n K minus a point is modeled on a Carnot group (see Subsection 3.3). Pansu defined a notion of differentiability in Carnot groups by using the Carnot homotheties to zoom in at a point (Definition 7.2). He proved a Rademacher-Stepanov type theorem for quasi-conformal homeomorphisms of Carnot groups (Theorem 7.3). This result, with a compactness property (Proposition 7.4), form together the core of the proof of Theorem 7.1.

Let (N, v) be a Carnot group and (δ t ) t∈R its Carnot homotheties (see Definition 3.6). We fix a scalar product on v and we propagate v to a N -invariant distribution in the tangent space of N . These data determine a Carnot-Carathéodory metric d N on N , which is N -invariant and multiplied by e t under δ t (see Subsection 3.4).

Definition 7.2 ([116]

). Let N, N be Carnot groups with Carnot homotheties δ t , δ t respectively. A map f :

N → N is δ-differentiable at z ∈ N if the maps w ∈ N → δ t (f (z) -1 f (zδ -t (w)) ∈ N
converge as t → +∞, uniformly on compact subsets of N , to a group homomorphism D z f : N → N , which commutes with δ t and δ t .

We equip N with its Haar measure (which coincides with the Hausdorff measure of (N, d N ), see for instance [START_REF] Hajlasz | Sobolev met Poincaré[END_REF]). The following version of Rademacher-Stepanov's theorem holds :

Theorem 7.3 ([116]). Every quasi-conformal homeomorphism f : (N, d N ) → (N , d N ) is δ-differentiable almost everywhere.
Recall from Subsection 3.3, that ∂H n K \ {∞} may be identified with a Carnot group (N, v). Its Lie algebra is n = K n-1 ⊕ K, with K central, and for x, y ∈ K n-1 :

[x, y] = {x, y}.

The horizontal space is v = K n-1 , and [v, v] = K (2 ). Let α be the associated derivation of n. The key point in the proof of Theorem 7.1, which distinguishes between the case R, C and the case Q, O, is the following compactness result : [START_REF] Pansu | Métriques de Carnot-Carathéodory et quasiisométries des espaces symétriques de rang un[END_REF]). For K = Q or O, the subgroup of Aut(n) which commutes with (e tα ) t∈R , is a semidirect product of a compact group with (e tα ) t∈R .

Proposition 7.4 ([
The following proof of Proposition 7.4 is rather different from [START_REF] Pansu | Métriques de Carnot-Carathéodory et quasiisométries des espaces symétriques de rang un[END_REF]. It relies on two lemmata. The first one, due to A. Kaplan and A. Tiraboschi, is of general interest.

Let n = v ⊕ z be a two-step nilpotent Lie algebra with center z. Let α be the associated derivation of n, and let H ⊂ Aut(n) be the subgroup that commutes with (e tα ) t∈R . It can be written as

H = { a 0 0 b | a ∈ GL(v), b ∈ GL(z), [ax, ay] = b([x, y])}. Let H 1 = { a 0 0 b ∈ H | det(a) = ±1} and H 0 = { a 0 0 b ∈ H 1 | b = id}.
We remark that H is the semidirect product of H 1 with (e tα ) t∈R , and that H 0 is closed and normal in H 1 . Moreover the group H 1 /H 0 is isomorphic to the image of H 1 by the projection map

π : a 0 0 b ∈ H → b ∈ GL(z).
Lemma 7.5 (Kaplan-Tiraboschi [START_REF] Kaplan | Automorphisms of non-singular nilpotent Lie algebras[END_REF]). Suppose that for every non-zero x ∈ v, the map adx : n → z is surjective. Then π(H 1 ) is a compact subgroup of GL(z).

Proof. We follow the proof in [START_REF] Kaplan | Automorphisms of non-singular nilpotent Lie algebras[END_REF]. Fix arbitrary scalar products on v and z. For x, y ∈ v and z ∈ z, the relation

x, T z y v = [x, y], z z
defines a linear map z → T z from z to End(v). By our hypothesis on adx, one has T z ∈ GL(v) for z = 0. Set P (z) := det(T z ).

It is a homogeneous polynomial that is nonzero on z \ {0}. Therefore its level sets are bounded.

Let now g = a 0 0 b ∈ H. Then we have

T b * z = a * T z a. Indeed, x, T b * z y v = [x, y], b * z z = b([x, y]), z z = [ax, ay], z z = ax, T z ay v =
x, a * T z ay v . Therefore P (b * z) = (deta) 2 P (z), and so P is π(H 1 ) * -invariant. Since the level sets of P are bounded, the group π(H 1 ) * is bounded in GL(z), and so is π(H 1 ). On the other hand, H 1 is a real algebraic group and π is a regular map. Thus π(H 1 ) is open in its closure (see [START_REF] Zimmer | Ergodic theory and semisimple groups, volume 81 of Monographs in Mathematics[END_REF] 3.1.3). Since we know from above that the closure of π(H 1 ) is compact, we obtain that π(H 1 ) is compact too. Lemma 7.6. Let n be a Lie algebra as in Proposition 7.4. Then the group H 0 is compact.

Proof. We prove it for Q, and leave the reader adapt the proof to the octonions. Let •, • be the scalar product on v defined by x, y = {x, y}. Let i, j, k be the standard basis of Q. For α = i, j, k, denote by ω α the symplectic form on v, so that [x, y] = ω i (x, y)i + ω j (x, y)j + ω k (x, y)k.

These forms are given by ω α (x, y) = -x, T α y , where T α is the right multiplication by α.

Let now a ∈ GL(v) such that [ax, ay] = [x, y]. For α = i, j, k, we have ω α (ax, ay) = ω α (x, y), and so a * T α a = T α .

This formula applied to a -1 yields by taking the inverses in both sides:

aT α a * = T α , because T -1 α = -T α .
It follows that aT i T j = T i (a * ) -1 T j = T i T j a. Thus a commutes with T i T j = -T k . Since we also have a * T k a = T k , we obtain that a * a = id. Thus a preserves the scalar product.

Proof of Proposition 7.4. From Lemma 7.5, the group H 1 /H 0 is compact. According to Lemma 7.6, H 0 is compact too. Thus H 1 is compact.

Z/(q i + 1)Z; to the vertice {i, i + 1}, the group Γ {i,i+1} := Γ {i} × Γ {i+1} . An abstract theorem of Haefliger [START_REF] Bridson | Metric spaces of non-positive curvature[END_REF] shows that the complex of groups is developable, i.e.

1. There exits a contractible 2-cell complex ∆, called the universal cover of the complex of groups; its 0 and 1-cells are labelled by the same symbols as above, and its 2-cells are isomorphic to the labeled complex R.

2. There exists a group Γ, called the fundamental group of the complex of groups. It acts geometrically and label-preserving on ∆, with Γ\∆ = R, and in such a way that the stabilizer in Γ of a 2-cell, or a 1-cell {i}, or a 0-cell {i, i + 1}, is isomorphic to the group Γ ∅ , Γ {i} , Γ {i,i+1} , respectively.

The group Γ admits the following presentation

Γ = s i , i ∈ Z/rZ | s qi+1 i = 1, [s i , s i+1 ] = 1 .
The link of a vertex {i, i + 1} of ∆ is the complete bipartite graph with (q i + 1) + (q i+1 + 1) vertices ( 3). Since this graph is a spherical building of girth 4, and since 4 × π/2 = 2π, one obtains that ∆ is a building. More precisely:

1. Its chambers are the 2-cells; its apartments are the subcomplexes that are isomorphic to the Coxeter tiling of H 2 R by copies of R. 2. Any pair of chambers is contained in an apartment.

3. If A 1 and A 2 are apartments whose intersection is not empty, there exists an isomorphism of labelled cell complexes φ :

A 1 → A 2 stabilizing pointwise A 1 ∩ A 2 .
We endow ∆ with the length metric induced by its 2-cells. Then ∆ is a CAT(-1)space. It enjoys the following properties:

• Its boundary is homeomorphic to the Menger curve [START_REF] Benakli | Polyèdres hyperboliques, passage du local au global[END_REF][START_REF] Kapovich | Hyperbolic groups with low-dimensional boundary[END_REF][START_REF] Dymara | Boundaries of right-angled hyperbolic buildings[END_REF].

• The group Isom(∆) is locally compact totally disconnected, and Γ is a cocompact lattice.

• The label-preserving isometries of ∆ form a non-linear, simple, finite index subgroup of Isom(∆) [START_REF] Haglund | Simplicité de groupes d'automorphismes d'espaces à courbure négative[END_REF].

Theorem 7.7 ( [START_REF] Bourdon | Rigidity of quasi-isometries for some hyperbolic buildings[END_REF]). Let ∆ 1 and ∆ 2 be right-angled Fuchsian buildings. Then any quasi-isometry F : ∆ 1 → ∆ 2 lies within bounded distance from an isometry.

As a consequence, the Mostow theorem and the Sullivan-Tukia theorem hold for the right-angled Fuchsian buildings too. Xie [START_REF] Xie | Quasi-isometric rigidity of Fuchsian buildings[END_REF] generalized Theorem 7.7 to all Fuchsian buildings that admit a geometric action.

The strategy of the proof is similar in spirit to the one of Theorem 7.1. One proves that there exists an H > 0 such that every quasi-Möbius homeomorphism f : ∂∆ 1 → ∂∆ 2 is H-quasi-conformal. The analytic ingredients consist of a Loewner metric on ∂∆ and some reasonable differential properties of quasi-Möbius homeomorphisms f : ∂∆ 1 → ∂∆ 2 . More precisely, we show that for almost every z ∈ ∂∆ 1 and every apartment A ⊂ ∆ 1 such that z ∈ ∂A, the limit

f ∂A (z) := lim w∈∂A w→z d(f (w), f (z)) d(w, z)
exists and belongs to (0, +∞). To prove that f is H-quasi-conformal for some uniform H, a key observation is that f ∂A (z) depends only on z and not on ∂A. This follows from that fact that for every pair of apartments A 1 , A 2 with z ∈ ∂A i , there exists an apartment A 3 whose boundary contains z locally coincides with ∂A 1 on the left side of z, and with ∂A 2 on the right side of z.

It is worth mentioning that, although ∆ is a CAT(-1)-space, the metric d CAT on ∂∆ is not Loewner. The Loewner metric on ∂∆ is associated to a chamber distance in ∆, see [START_REF] Bourdon | Poincaré inequalities and quasiconformal structure on the boundary of some hyperbolic buildings[END_REF][START_REF] Bourdon | Rigidity of quasi-isometries for some hyperbolic buildings[END_REF].

Futher results

1) Kapovich and Kleiner [START_REF] Kapovich | Hyperbolic groups with low-dimensional boundary[END_REF] gave examples of hyperbolic groups Γ with connected boundary such that Homeo(∂Γ) = Γ. Therefore any quasi-isometry of Γ lies within bounded distance from the left multiplication by a g ∈ Γ. This situation contrasts with the case of rank-one symmetric spaces and Fuchsian buildings, where the homeomorphim group of the boundary is infinite dimensional.

2) Two subgroups Γ 1 and Γ 2 of a group G are said to be commensurable if there is g ∈ G such that gΓ 1 g -1 ∩Γ 2 is of finite index in gΓ 1 g -1 and in Γ 2 . F. Haglund [START_REF] Haglund | Commensurability and separability of quasiconvex subgroups[END_REF] worked out the commensurability question of cocompact lattices in the isometry group of certain negatively curved simplicial complexes. In particular, for rightangled Fuchsian buildings ∆ (with r ≥ 5 and q i = q ≥ 3) he showed that all cocompact lattices in Isom(∆) are commensurable and linear.

Note that commensurability and linearity fail for non-cocompact lattices in Isom(∆) [START_REF] Thomas | Existence, covolumes and infinite generation of lattices for Davis complexes[END_REF][START_REF] Rémy | Topological simplicity, commensurator super-rigidity and nonlinearities of Kac-Moody groups[END_REF]. For a survey on recent developments on buildings and groups, see for instance [START_REF] Rémy | On some recent developments in the theory of buildings. Proceedings of the ICM Seoul[END_REF].

3) (4 ) By Heintze's theorem [START_REF] Heintze | On homogeneous manifolds of negative curvature[END_REF], every negatively curved homogeneous manifold is isometric to a solvable Lie group (with a certain left invariant Riemannian metric) of the form G = N R, where N is a simply connected nilpotent Lie group , and R acts on N by expanding automorphisms. More precisely, the R-action is given by e tα , where α is a derivation of the Lie algebra n of N whose eigenvalues all have positive real parts. Such groups G are called Heintze groups. When all eigenvalues are real, G is said to be purely real. It is known that every Heintze group is bi-Lipschitz homeomorphic to a purely real Heintze group [START_REF] Cornulier | On the quasi-isometric classification of locally compact groups[END_REF]. A Heintze group G is of Carnot type, if it is bi-Lipschitz homeomorphic to N α R, where N is a Carnot group and α is the derivation associated to the Carnot decomposition of n, see Definition 3.6. Observe that the boundary at infinity of every Heintze group G is identified canonically with the one point compactification N ∪ {∞}.

There are three major conjectures about Heintze groups G:

Conjecture 7.8 (Pointed sphere conjecture [START_REF] Cornulier | On the quasi-isometric classification of locally compact groups[END_REF]). Any quasi-isometry of G stabilizes ∞ unless G is bi-Lipschitz homeomorphic to a rank-one symmetric space.

Conjecture 7.9 (Quasi-isometric classification [START_REF] Hamenstädt | Theorie von Carnot-Carathéodory Metriken und ihren Anwendungen[END_REF][START_REF] Cornulier | On the quasi-isometric classification of locally compact groups[END_REF]). Let G 1 and G 2 be purely real Heintze groups. If they are quasi-isometric, then they are isomorphic.

Conjecture 7.10 (Quasi-isometric rigidity [START_REF] Xie | Quasiisometries of negatively curved homogeneous manifolds associated with Heisenberg groups[END_REF]). Any quasi-isometry F : G 1 → G 2 between Heintze groups lies at bounded distance from an almost similarity (5 ) unless one of the groups is bi-Lipschitz homeomorphic to a symmetric space.

Conjecture 7.8 was proved by Pansu [START_REF] Pansu | Dimension conforme et sphère à l'infini des variétés à courbure négative[END_REF] for the non-Carnot type Heintze groups with α diagonalizable over C. It was generalized by Carrasco [START_REF] Piaggio | Orlicz spaces and the large scale geometry of Heintze groups[END_REF] to all non-Carnot type Heintze groups. Conjecture 7.9 was established by Pansu [START_REF] Pansu | Métriques de Carnot-Carathéodory et quasiisométries des espaces symétriques de rang un[END_REF] for the Carnot type Heintze groups, by using Theorem 7.3. The three conjectures hold when N is abelian (Shanmugalingam-Xie [START_REF] Shanmugalingam | A rigidity property of some negatively curved solvable Lie groups[END_REF] and Xie [START_REF] Xie | Large scale geometry of negatively curved R n R[END_REF]). They also hold when N is a Heisenberg group and α is diagonalizable over R [START_REF] Xie | Quasiisometries of negatively curved homogeneous manifolds associated with Heisenberg groups[END_REF]. Conjecture 7.10 was proved by Carrasco [START_REF] Piaggio | Orlicz spaces and the large scale geometry of Heintze groups[END_REF] for the non-Carnot type Heintze groups, and by Le Donne and Xie [START_REF] Le Donne | Rigidity of fiber-preserving quasisymmetric maps[END_REF] for the reducible ( 6) Carnot type Heintze groups.

Some recent developments and perspectives

This section reports on recent progress on some open problems about quasiconformal geometry of group boundaries. We discuss the Cannon conjecture in Subsection 8.1 and the combinatorial Loewner property in Subsection 8.2. As usual the section ends with a survey of several related results.

Cannon conjecture

A major problem in geometric group theory is the following conjecture. Conjecture 8.1 (Cannon's conjecture [START_REF] Cannon | Recognizing constant curvature discrete groups in dimension 3[END_REF]). Suppose Γ is a hyperbolic group whose boundary is homeomorphic to the Euclidean 2-sphere, then Γ acts geometrically on H 3 R . Historically, this conjecture was motivated by Thurston's hyperbolization conjecture (recently solved by Perelman): Every closed, aspherical, irreducible, atoroidal 3-manifold admits a Riemannian metric of constant curvature -1. Cannon's conjecture also provides an approach to solve an open problem due to Wall: Is every 3-dimensional Poincaré duality group a 3-manifold group? Sullivan-Tukia rigidity (Theorem 6.2) implies that Cannon's conjecture is equivalent to: Conjecture 8.2 (Uniformization conjecture). Suppose that Γ is a hyperbolic group whose boundary is homeomorphic to the Euclidean sphere S 2 , then ∂Γ is quasi-Möbius homeomorphic to S 2 . Hence Cannon's conjecture reduces to a problem of quasi-conformal geometry. We remark that the analogous problem for the circle is solved, since every approximately self-similar metric circle is quasi-Möbius homeomorphic to the Euclidean one [START_REF] Tukia | Quasisymmetric embeddings of metric spaces[END_REF]. It follows that the analog of Cannon's conjecture for H We describe in the sequel an approach to Conjecture 8.2 based on the so-called combinatorial modulus -a combinatorial variant of the analytic modulus defined in Subsection 4.3. Versions of the combinatorial modulus have been considered by several authors in connection with the Cannon conjecture [START_REF] Cannon | The combinatorial Riemann mapping theorem[END_REF][START_REF] Cannon | Recognizing constant curvature discrete groups in dimension 3[END_REF][START_REF] Cannon | Sufficiently rich families of planar rings[END_REF][START_REF] Bonk | Quasisymmetric parametrizations of twodimensional metric spheres[END_REF][START_REF] Haissinsky | Empilements de cercles et modules combinatoires[END_REF], and in a more general context [START_REF] Pansu | Dimension conforme et sphère à l'infini des variétés à courbure négative[END_REF][START_REF] Tyson | Quasiconformality and quasisymmetry in metric measure spaces[END_REF]. Our presentation follows [START_REF] Bourdon | Combinatorial modulus, the combinatorial Loewner property, and Coxeter groups[END_REF].

Let Z be a compact metric space, let k ∈ N, and let κ ≥ 1. A finite graph G k is called a κ-approximation of Z on scale k, if it is the incidence graph of a covering of Z, such that for every vertex v ∈ G 0 k there exists z v ∈ Z with

B(z v , κ -1 2 -k ) ⊂ v ⊂ B(z v , κ2 -k ),
and for v, w ∈ G 0 k with v = w:

B(z v , κ -1 2 -k ) ∩ B(z w , κ -1 2 -k ) = ∅.
Note that we identify every vertex v of G k with the corresponding subset in Z. A collection of graphs {G k } k∈N is called a κ-approximation of Z, if for each k ∈ N the graph G k is a κ-approximation of Z on scale k. Let γ ⊂ Z be a continuous curve and let ρ :

G 0 k → R + be any function. The ρ-length of γ is L ρ (γ) = v∩γ =∅ ρ(v). For p ≥ 1 the p-mass of ρ is M p (ρ) = v∈G 0 k ρ(v) p .
Let F be a non-empty family of (continuous) curves in Z. We define the G kcombinatorial p-modulus of F by

Mod p (F, G k ) = inf ρ M p (ρ),
where the infimum is over all F-admissible functions, i.e. functions ρ : G 0 k → R + which satisfy L ρ (γ) ≥ 1 for every γ ∈ F. It enjoys the following properties 1. The function Mod p (•, G k ) is non-decreasing and finitely subadditive.

If every curve in

F 2 contains a curve in F 1 , then one has Mod p (F 2 , G k ) ≤ Mod p (F 1 , G k ).
3. When Z is a doubling metric space ( 7) the combinatorial modulus does not depend on the choice of the graph approximation, up to a multiplicative constant.

4. If Z 1 is quasi-Möbius homeomorphic to a p-regular metric space Z 2 , then the combinatorial p-modulus on Z 1 is comparable to the analytic p-modulus on Z 2 [START_REF] Haissinsky | Empilements de cercles et modules combinatoires[END_REF].

The following quasi-Möbius characterization of the Euclidean 2-sphere appears in [START_REF] Bourdon | Combinatorial modulus, the combinatorial Loewner property, and Coxeter groups[END_REF]. It is a straightforward application of techniques and results developed in [START_REF] Cannon | Sufficiently rich families of planar rings[END_REF][START_REF] Bonk | Quasisymmetric parametrizations of twodimensional metric spheres[END_REF]. Theorem 8.3. Suppose that Z is an approximately self-similar metric 2-sphere. Let {G k } k∈N be a κ-approximation of Z. For d 0 > 0 denote by F 0 the set of curves γ ⊂ Z with diam(γ) ≥ d 0 . Then Z is quasi-Möbius homeomorphic to the Euclidean 2-sphere, if and only if, for every d 0 > 0 small enough, there exists a constant C ≥ 1 such that for every k ∈ N one has

Mod 2 (F 0 , G k ) ≤ C .

Sketch of proof.

The direct implication is a consequence of Property 4 above. Our proof of the reverse implication combines several arguments from [START_REF] Bonk | Quasisymmetric parametrizations of twodimensional metric spheres[END_REF]. First, selfsimilarity allows one to improve the modulus control assumption as follows. There exists a positive increasing function ψ of (0, +∞) with lim t→0 ψ(t) = 0, such that for every pair of disjoint non-degenerate compact connected subsets A, B ⊂ Z and every integer k satisfying 2 -k ≤ min{diam(A), diam(B)}, one has

Mod 2 (A, B, G k ) ≤ ψ(∆(A, B) -1 ), (8.4)
where ∆(A, B) denotes the relative distance, defined in (4.5). Secondly, Z being an approximately self-similar compact manifold, it is doubling and linearly locally contractible i.e. there exists a constant λ ≥ 1 such that every ball B(z, r) ⊂ Z with 0 < r < diam(Z)/λ is contractible in B(z, λr). Bonk and Kleiner use the last properties to construct a κ-approximation {G k } k∈N of Z such that each graph G k is (essentially) homeomorphic to the 1-skeleton of a triangulation of the 2-sphere. Therefore, according to the Andreev-Koebe-Thurston theorem, G k is the incidence graph of a circle packing in the Euclidean sphere S 2 (unique up to a homography). For every k

∈ N and v ∈ G 0 k , let z v ∈ Z be such that B(z v , κ -1 2 -k ) ⊂ v ⊂ B(z v , κ2 -k ). We obtain a map f k : {z v | v ∈ G 0 k } → S 2
which sends z v to the center of the corresponding disc. Now the f k 's, when appropriately normalized, form an equicontinuous sequence that converges to a homeomorphism f : Z → S 2 , which satisfies for every pair of disjoint non-degenerate compact connected subsets A, B ⊂ Z:

∆(f (A), f (B)) ≥ η(∆(A, B)), (8.5)
where η is a positive function of (0, +∞) that satisfies lim t→+∞ η(t) = 0. To prove these properties, one uses the modulus bounds (8.4), and relates the modulus associated to circle packings with the analytic modulus of S 2 . Finally, inequality (8.5) implies that f is quasi-Möbius (Lemmata 2.10 and 3.3 in [START_REF] Bonk | Quasisymmetric parametrizations of twodimensional metric spheres[END_REF]).

Theorem 8.3 provides a strategy to prove Cannon's conjecture. Let Γ be a 2-sphere boundary hyperbolic group. To show that ∂Γ satisfies the assumptions of Theorem 8.3, one can try to use the action of Γ on ∂Γ. A crucial observation is the following. Say that two curve families F, G in ∂Γ cross, if every curve in F crosses every curve in G. When F, G cross, then one has:

Mod 2 (F, G k ) • Mod 2 (G, G k ) ≤ 1. (8.6)
This is indeed a classical property of the analytic modulus in the Euclidean 2sphere. It generalizes to metric 2-spheres and combinatorial 2-modulus [START_REF] Cannon | Sufficiently rich families of planar rings[END_REF][START_REF] Haissinsky | Empilements de cercles et modules combinatoires[END_REF]. Now, suppose that we are given a curve family F in ∂Γ and g ∈ Γ such that F and g(F) cross. Since Mod 2 (•, G k ) is invariant by bi-Lipschitz homeomorphism up to a multiplicative constant, we obtain from (8.6) that Mod 2 (F, G k ) is bounded by above independently of k.

This strategy, in a more elaborate form, can be applied to Coxeter groups. We obtain ( 8 ): Theorem 8.7 ( [START_REF] Bourdon | Combinatorial modulus, the combinatorial Loewner property, and Coxeter groups[END_REF]). Let Γ be a hyperbolic Coxeter group whose boundary is homeomorphic to the 2-sphere. Then Γ acts geometrically on H 3 R .

Combinatorial Loewner property

Most of the rigidity results discussed so far rely on the Loewner property of the boundary. Unfortunately, among the currently known examples of Loewner spaces, the only ones which arise as boundaries of hyperbolic groups are the boundaries of rank-one symmetric spaces and Fuchsian buildings. In order to make a step toward improving this situation, Kleiner [START_REF] Kleiner | The asymptotic geometry of negatively curved spaces: uniformization, geometrization and rigidity[END_REF] introduced a combinatorial variant called the combinatorial Loewner property.

Suppose thet Z is an arcwise connected compact doubling metric space. Let {G k } k∈N be a κ-approximation of Z. Denote by F(A, B) the family of curves 8 M. Davis pointed out to the author that Theorem 8.7 can be also established as follow. A theorem of Bestvina-Mess [START_REF] Bestvina | The boundary of negatively curved groups[END_REF] and the boundary hypothesis show that Γ is a virtual 3-dimensional Poincaré duality group. Then Theorem 10.9.2 of [START_REF] Davis | The geometry and topology of Coxeter groups[END_REF] implies that Γ decomposes as Γ = Γ 0 × Γ 1 , where Γ 0 is a finite Coxeter group and where Γ 1 is a Coxeter group whose nerve is a 2-sphere. By applying Andreev's theorem to the dual polyhedron to the nerve, one obtains that Γ 1 acts on H 3 R as a cocompact reflection group.

joining two subsets A, B ⊂ Z and by Mod p (A, B, G k ) its G k -combinatorial pmodulus. Recall that ∆(A, B) denotes the relative distance defined in (4.5).

Definition 8.8. Suppose that p > 1. Then Z satisfies the combinatorial p-Loewner property if there exist two positive increasing functions φ, ψ on (0, +∞) with lim t→0 ψ(t) = 0, such that 1. For every pair of disjoint compact connected subsets A, B ⊂ Z, and every k with 2 -k ≤ min{diam A, diam B}, one has: φ(∆(A, B) -1 ) ≤ Mod p (A, B, G k ).

2. For every pair of concentric balls B(z, r), B(z, R) ⊂ Z with 0 < 2r ≤ R, and every k with 2 -k ≤ r, one has:

Mod p (B(z, r), Z \ B(z, R), G k ) ≤ ψ(r/R).

We say that Z satisfies the combinatorial Loewner property (CLP) if it satisfies the combinatorial p-Loewner property for some p > 1.

Theorem 8.9 ([30]).

1. If Z is a compact Ahlfors p-regular Loewner space, then Z satisfies the combinatorial p-Loewner property.

2. If Z 1 satisfies the CLP and Z 2 is quasi-Möbius homeomorphic to Z 1 , then Z 2 also satisfies the CLP (with the same exponent).

Conversely, Kleiner made the following conjecture.

Conjecture 8.10 ( [START_REF] Kleiner | The asymptotic geometry of negatively curved spaces: uniformization, geometrization and rigidity[END_REF]). If Z satisfies the CLP and is approximately self-similar, then Z is quasi-Möbius homeomorphic to a Loewner space.

Since the CLP is invariant under quasi-Möbius homeomorphisms -unlike the Loewner property (see Theorem 4.7) -it is in principle easier to verify that a given metric space admits the CLP, than to prove that it is quasi-Möbius homeomorphic to a Loewner space.

In addition to the already known Loewner spaces, examples of metric spaces that satisfy the CLP include the standard Sierpinski carpet and Menger curve [START_REF] Bourdon | Combinatorial modulus, the combinatorial Loewner property, and Coxeter groups[END_REF], the boundaries of some hyperbolic Coxeter groups [START_REF] Bourdon | Combinatorial modulus, the combinatorial Loewner property, and Coxeter groups[END_REF], and the boundaries of some hyperbolic buildings of dimension 3 and 4 [START_REF] Clais | Combinatorial modulus on boundary of right-angled hyperbolic buildings[END_REF]. See also [START_REF] Bourdon | Some applications of p -cohomology to boundaries of Gromov hyperbolic spaces[END_REF] for examples of hyperbolic group boundaries which do not satisfy the CLP.

Further results

1) Gromov and Thurston [START_REF] Gromov | Pinching constants for hyperbolic manifolds[END_REF] constructed, for every n ≥ 4, examples of compact n-manifolds whose sectional curvatures are arbitrarly close to -1, but whose universal covers are not quasi-isometric to any symmetric space. Y. Benoist [START_REF] Benoist | Convexes hyperboliques et quasiisométries[END_REF] constructed examples of compact locally CAT(-1) 4-manifold, whose universal cover is not quasi-isometric to any symmetric spaces, but whose fundamental groups admit a properly discontinuous cocompact projective action on a strictly convex open subset of the projective space P 4 R . 2) Bonk and Kleiner [START_REF] Bonk | Quasisymmetric parametrizations of twodimensional metric spheres[END_REF] obtained several quasi-Möbius characterizations of the Euclidean sphere S 2 . In particular • Let Z be an Ahlfors 2-regular metric space homeomorphic to S 2 . Then Z is quasi-Möbius to S 2 if and only if Z is linearly locally contractible (see the definition in the proof of Theorem 8.3).

• Let Q ≥ 2 and Z be an Ahlfors Q-regular metric space homeomorphic to S 2 . If Z is Loewner, then Q = 2 and Z is quasi-Möbius to S 2 .

The techniques of proof are those at the origin of Theorem 8.3.

3) As a consequence of Theorem 4.7, the Hausdorff dimension of a Loewner space is minimal among all quasi-Möbius homeomorphic Ahlfors-regular metric spaces. Bonk and Kleiner [START_REF] Bonk | Conformal dimension and Gromov hyperbolic groups with 2-sphere boundary[END_REF] established a converse statement for boundaries of hyperbolic groups. Let Q > 1 and let Z be an Ahlfors Q-regular metric space quasi-Möbius homeomorphic to the boundary of a hyperbolic group. Suppose Q is minimal among all Ahlfors-regular metric spaces quasi-Möbius homeomorphic to Z. Then Z is a Loewner space.

4) Kapovich and Kleiner [START_REF] Kapovich | Hyperbolic groups with low-dimensional boundary[END_REF] conjectured that every hyperbolic group whose boundary is homeomorphic to the Sierpinski carpet, is virtually the fundamental group of a compact 3-manifold with constant curvature -1 and non-empty totally geodesic boundary. They observed that this conjecture is implied by Cannon's conjecture. Bonk and Kleiner [START_REF] Bonk | Quasiconformal geometry of fractals[END_REF] announced the Kapovich-Kleiner conjecture under the additional assumption that the boundary is quasi-Möbius homeomorphic to a Qregular metric space with Q < 2.

Haïssinsky [START_REF] Haissinsky | Hyperbolic groups with planar boundaries[END_REF] proved more generally that a hyperbolic group whose boundary is planar and quasi-Möbius homeomorphic to a Q-regular metric space with Q < 2, acts quasi-convex geometrically on H 3 R . In [START_REF] Bourdon | Combinatorial modulus, the combinatorial Loewner property, and Coxeter groups[END_REF] the Kapovich-Kleiner conjecture is established for Coxeter groups, as a consequence of Theorem 8.7. [START_REF] Markovic | Criterion for Cannon's conjecture[END_REF] proved that Cannon's conjecture holds if one assumes in addition that every two distint points in ∂Γ can be separated by the limit set of a quasi-convex surface subgroup of Γ. Haïssinsky [START_REF] Haissinsky | Hyperbolic groups with planar boundaries[END_REF] proved more generally that a hyperbolic group, whose boundary is planar and satisfies the above separation assumption, acts quasi-convex geometrically on H 3 R . 5) In addition to hyperbolic group boundaries, we remark that discrete quasiconformal techniques apply to several other dynamical situations. This includes finite subdivision rules [START_REF] Cannon | Finite subdivision rules[END_REF][START_REF] Cannon | Constructing rational maps from subdivision rules[END_REF], rational maps and ramified covers [START_REF] Bonk | Quasiconformal geometry of fractals[END_REF][START_REF] Haissinsky | Coarse expanding conformal dynamics[END_REF][START_REF] Meyer | Invariant Peano curves of expanding Thurston maps[END_REF][START_REF] Haissinsky | Minimal Ahlfors regular conformal dimension of coarse conformal dynamics on the sphere[END_REF][START_REF] Bonk | Expanding Thurston maps[END_REF], quasiconformal geometry on fractals like the Sierpinski carpets [START_REF] Bonk | Quasiconformal geometry of fractals[END_REF][START_REF] Bonk | Rigidity of Schottky sets[END_REF][START_REF] Bonk | Uniformization of sierpiǹski carpets in the plane[END_REF][START_REF] Bonk | Quasisymmetric rigidity of square Sierpiǹski carpets[END_REF][START_REF] Haissinsky | Quasi-Moebius group actions[END_REF].

5) Markovic

  b 1 , b 2 , b 3 are pairwise distinct then the previous lemma applies. Suppose b 1 = b 2 and write it b for simplicity. Let c ∈ Z 2 \ {b} and consider

3 .

 3 f is absolutely continuous along almost curves γ : [a, b] → Z 1 . In other words, for γ lying outside of a set of curves of zero modulus, the maps f • γ are absolutely continuous with respect to 1-dimensional Hausdorff measure in the target.

  The associated cross-ratio (see Definition 2.11) satisfies [z 1 , z 2 , z 3 , z 4 ] = exp(z 1 |z 2 |z 3 |z 4 ). The metric d CAT on its boundary can be describe as follows. In the ball model B n ⊂ R n of H n R , choose the origin O to be the center of B n . Then the metric d CAT on ∂H n R is half of the chordal distance on the boundary sphere of B n . A description of d CAT for the other rank-one symmetric spaces is given in Proposition 3.7.

	Example 2.18. Let H n R be the n-dimensional real hyperbolic space of constant
	curvature -1.

  Proposition 3.7. The metrics d CAT and d CC on ∂H n K are Lipschitz equivalent. Their Hausdorff dimension is kn + k -2. Proof. Observe that d CAT and d CC are both invariant under the isotropy group of O. Since this group acts transitively on ∂H n K , it is enough to prove the existence of a point z 0

  2 R holds (see the discussion after Theorem 6.2). On the other hand, there is no analog of Cannon's conjecture for H n R when n ≥ 4. For example ∂H 2 C and ∂H 4

	R are topological R are not quasi-isometric (Corollary 4.8). See subsection C and H 4 3-spheres, but H 2
	8.3 for more examples.

This convention is only relevant when K is non abelian. It allows M (n + 1, K) to act linearly on K n+1 by multiplication on the left.

These expressions have been established in Subsection

3.3 for K = R, C, Q. They are also valid when K = O, see[START_REF] Mostow | Strong rigidity of locally symmetric spaces[END_REF] p. 141.

The link of a vertex v is the graph whose vertices are the edges of ∆ that contain v, and whose edges are the pair of edges of ∆ that are contained in a 2-cell.

I owe Matias Carrasco several explanations about the material in this paragraph.

An almost similarity is a quasi-isometry whose left and right handside multiplicative constants are equal.

[START_REF] Besson | Entropies et rigidités des espaces localement symétriques de courbure strictement négative[END_REF] A Carnot group (N, v) is reducible if the horizontal subspace v contains a proper non-trivial subspace that is invariant by the graded automorphisms of (N, v).

A metric space is said to be doubling if there is a constant n ∈ N such that every ball B(z, r) can be covered by at most n balls of radius r/2. For instance Ahlfors-regular metric spaces (Definition 2.13) are doubling.

Acknowledgements

Parts of the paper are issued from minicourses I taught. I am thankful to their

Supported in part by ANR project "GdSous" and by Labex Cempi.

of pairwise distinct points of ∂Γ. Thus f Γf -1 acts properly discontinuously and cocompactly on the space of triples of parwise distinct points of S n-1 . Therefore it is a cocompact lattice in Isom(H n R ), thanks to Proposition 2.15.

6.3 Another proof of Mostow's theorem for H n R Mostow's theorem is stated in Subsection 5.1. The following proof (for K = R) is based on the zoom method. It is due to Tukia [START_REF] Tukia | Differentiability and rigidity of Möbius groups[END_REF]. It appears also in [START_REF] Kapovich | Hyperbolic manifolds and discrete groups[END_REF].

be the induced ϕ-equivariant quasi-conformal homeomorphism. By Rademacher-Stepanov's theorem (see e.g. [START_REF]Lectures on n-dimensional quasiconformal mappings[END_REF]) it is differentiable a.e. Since quasi-conformal homeomorphisms that are 1-quasiconformal a.e. are conformal diffeomorphims (see e.g. [START_REF] Tukia | A remark on 1-quasiconformal maps[END_REF]), and since Conf(S n-1 ) = Moeb(S n-1 ) = Isom(H n R ) by Liouville's theorem, it is enough to prove that D z f is a conformal linear map for a.e. z ∈ S n-1 .

Suppose f is differentiable at z 0 ∈ S n-1 and set L := D z0 f . In the upper half-space model of H n R , let 0 = (0, ..., 0) and O = (0, ..., 0, 1). We can assume that

stabilizes the constant field of ellipsoids equal to L(S n-2 ) (measurable fields of ellipsoids are defined in Subsection 6.2).

Suppose by contradiction that L is not conformal. Then the linear subspace generated by the largest axes of L(S n-2 ) is a proper non-zero subspace

fixes ∞, which contradicts the fact that it is a lattice in Isom(H n R ).

Further results

1) The zoom method is a main ingredient in the proof of the following result by R. Schwartz [START_REF] Schwartz | The quasi-isometric classification of rank one lattices[END_REF]. Let S 1 and S 2 be rank-one non-compact symmetric spaces different from H 2 R . Let Γ 1 and Γ 2 be non-cocompact lattices in Isom(S 1 ) and Let G ⊂ Homeo(∂S) be the group of quasi-Möbius homeomorphisms. Theorem 7.3 and Proposition 7.4 show that there exists a H ≥ 1 such that every g ∈ G is a.e. H-quasi-conformal. This implies -because ∂S is Loewner -that G is a uniform quasi-conformal group (see Theorems 4.7 and 4.10).

Let Γ ⊂ Isom(S) be a cocompact lattice. It follows from above and from Subsection 2.4, that G and Γ satisfy the assumptions of Corollary 4.11. Therefore G ⊂ Moeb(∂S, d CAT ) = Isom(S). Theorem 2.12 completes the proof.

Pansu's original proof differs a little bit at the end from the one above. His argument is the following. Theorem 7.3 and Proposition 7.4 imply that every g ∈ G is a 1-quasi-conformal homeomorphism of (∂S, d CC ). Pansu proves that every 1-quasi-conformal homeomorphism f of N is the boundary extension of an isometry of S. To do so, he shows that f can by written as a Carnot homothety composed with an isometry of N . The argument is based on the fact that 1quasi-conformal homeomorphisms preserve moduli of curves (see subsection 4.3). To complete the proof, it remains to show that the isometries of N that fix the identity lie in Aut(N ), and that Aut(N ) ⊂ Isom(S).

Right-angled Fuchsian buildings

Another family of hyperbolic spaces for which rigidity of quasi-isometries holds is provided by the so-called Fuchsian buildings. Fuchsian buildings are Tits buildings whose apartments are isomorphic to a Coxeter tiling of H 2 R . For simplicity we will consider only the right-angled ones. We refer to [START_REF] Davis | The geometry and topology of Coxeter groups[END_REF] and its references for Tits buildings, and to [START_REF] Bourdon | Immeubles hyperboliques, dimension conforme et rigidité de Mostow[END_REF][START_REF] Bourdon | Rigidity of quasi-isometries for some hyperbolic buildings[END_REF] for right-angled Fuchsian buildings.

A nice way to define right-angled Fuchsian buildings uses complexes of groups. Let r ≥ 5 be an integer, let R be a regular right angled r-gon in H 2 R , and let (q 1 , ..., q r ) be an r-tuple of integers with all q i ≥ 2. We label clockwise the edges of R by {1}, ..., {r}, and its vertices by {1, 2}, ..., {r -1, r}, {r, 1} in a way compatible with the adjacence relation. Then we define a complex of groups as follows. To the face of R, we attach the group Γ ∅ := {1}; to the edge {i}, the group Γ {i} :=