Let M=M(g) be the mapping class group of a surface of genus g > 1 (resp. M=Aut(F_g) the automorphism group of the Free group on g generators ). As it is well known, M is mapped onto the symplectic group Sp(2g,Z) (resp. the general linear group GL(g,Z) ). We will show that this is only a first case in a series: in fact, for every pair (S,r) when S is a finite group with less than g generators and r is a Q-irreducible representation of S, we associate an arithmetic group which is then shown to be a virtual quotient of M. The case when S is the trivial group gives the above Sp(2g,Z) ( resp. GL(g,Z) ) but many new quotients are obtained. For example it is used to show that M(2) (resp. Aut(F_3) ) is virtually mapped onto a non-abelian free group. Another application is an answer to a question of Kowalski: generic elements in the Torelli groups are hyperbolic and fully irreducible. Joint work with Fritz Gruenwald, Michael Larsen and Justin Malestein .

• Thematic program: Fellows of the Institut CNRS Pauli 2015/2016 (fellows 2015/2016)
• Event: Workshop on "Boundaries of groups and representations" (2016)

A theorem of Thurston says that if M is a finite volume non-compact hyperbolic manifold of dimension 3 (say with one cusp to simplify) then the manifold of dimension 3 obtained by filling (Dehn filling) the cusp is hyperbolic except in a finite number of cases. The hyperbolization of finite volume non-compact orbifold is possible only in dimension 2 or 3. We will exhibit examples of hyperbolic polytopes of dimension 4 which admit a projectivization of their Dehn filling. During this talk, "projectivize" will mean realise as the quotient of a properly convex open set of the real projective space by a discrete subgroup of projective transformation (preserving the convex). This is a joint work with Suhyoung Choi (KAIST) and Gye-Seon Lee (Heidelberg).

• Thematic program: Fellows of the Institut CNRS Pauli 2015/2016 (fellows 2015/2016)
• Event: Workshop on "Boundaries of groups and representations" (2016)

This is joint work with Antoine Clais (Technion). The combinatorial Loewner property (CLP) is a property of metric spaces invariant under quasi-Moebius homeomorphisms. It has been introduced by M. Bonk and B. Kleiner as a combinatorial counterpart of the classical Loewner property. Conjecturally, Gromov group boundaries satisfying the CLP are quasi-Moebius homeomorphic to Loewner spaces. For Loewner boundaries various quasi-conformal analysis techniques have been developed in order to achieve rigidity results. Not many group boundaries with the CLP are known. We present new classes of Gromov boundaries, in dimensions greater than one, satisfying the CLP. The underlying groups are hyperbolic right-angled Coxeter groups and lattices in associated buildings.

• Thematic program: Fellows of the Institut CNRS Pauli 2015/2016 (fellows 2015/2016)
• Event: Workshop on "Boundaries of groups and representations" (2016)

There is a classical result first due to Keen known as the collar lemma for hyperbolic surfaces. A consequence of the collar lemma is that if two closed curves A and B on a closed orientable hyperbolizable surface have non-zero geometric intersection number, then there is an explicit lower bound for the length of A in terms of the length of B, which holds for any hyperbolic structure on the surface. By slightly weakening this lower bound, we generalize this statement to hold for all Hitchin representations. Joint work with Tengren Zhang.

• Thematic program: Fellows of the Institut CNRS Pauli 2015/2016 (fellows 2015/2016)
• Event: Workshop on "Boundaries of groups and representations" (2016)

In this talk I give a definition of Anosov representation that does not use geodesic flow. Then I give a characterization in terms of coarse geometry of the orbit map in the symmetric space. This leads to the notion of Morse subgroups and to a Morse lemma for higher rank symmetric spaces. This is joint work with B. Leeb and M. Kapovich.

• Thematic program: Fellows of the Institut CNRS Pauli 2015/2016 (fellows 2015/2016)
• Event: Workshop on "Boundaries of groups and representations" (2016)

We give a geometric interpretation of the maximal Satake compactification of symmetric spaces X=G/K of noncompact type, showing that it arises by attaching the horofunction boundary for a suitable G-invariant "polyhedral" Finsler metric on X. We then discuss the topological dynamics of discrete subgroups Gamma"<"G on this compactification. We show that there exist natural domains of proper discontinuity for Gamma extending X, and that the Gamma-action on these domains is cocompact if Gamma is an Anosov subgroup. This leads to natural bordifications resp compactifications of the locally symmetric spaces X/Gamma as orbifolds with corners by attaching quotients of domains of discontinuity at infinity. This is joint work with Misha Kapovich.

• Thematic program: Fellows of the Institut CNRS Pauli 2015/2016 (fellows 2015/2016)
• Event: Workshop on "Boundaries of groups and representations" (2016)

When is a lattice in a Euclidean building linear? We will explain that answers to that question can be obtained by combining tools of various origins: ergodic theory, structure theory of disconnected locally compact groups, and classical theory of projective planes. Based on joint work with Uri Bader and Jean Lécureux.

• Thematic program: Fellows of the Institut CNRS Pauli 2015/2016 (fellows 2015/2016)
• Event: Workshop on "Boundaries of groups and representations" (2016)

We prove that any right-angled Coxeter group on k generators admits a proper affine action on R^{k(k-1)/2}. This yields proper affine actions for many other groups, including all Coxeter groups. Joint work with J. Danciger and F. Guéritaud.

• Thematic program: Fellows of the Institut CNRS Pauli 2015/2016 (fellows 2015/2016)
• Event: Workshop on "Boundaries of groups and representations" (2016)

Jointly with Anna Wienhard, we obtain a better understanding of the compact $\mathbf{R}\mathbb{P}^{2n-1}$-manifolds coming from maximal representations into the symplectic group $\mathrm{Sp}(2n, \mathbf{R}$, and in particular of their topology. This is based on the special properties of the boundary map into the Lagrangian variety.

• Thematic program: Fellows of the Institut CNRS Pauli 2015/2016 (fellows 2015/2016)
• Event: Workshop on "Boundaries of groups and representations" (2016)

In this talk I will describe the stable and unstable leaves for the geodesic flow on the space of non-wandering space like geodesics of a Margulis Spacetime. I will also describe how monodromy of Margulis Spacetimes are “Anosov representations in non semi-simple Lie groups”. Finally using the Anosov property I will define the Pressure metric on the Moduli Space of Margulis Spacetimes and discuss some of its properties.

• Thematic program: Fellows of the Institut CNRS Pauli 2015/2016 (fellows 2015/2016)
• Event: Workshop on "Boundaries of groups and representations" (2016)

A group is called $N$]uniformly simple if for every nontrivial conjugacy class $C$, $(C^\pm)^{\leq N}$ covers the whole group. Every uniformly simple group is simple. It is known that many group with geometric or dynamical origin are simple. In the talk we prove that, in fact, many of them are uniformly simple. The result are due to the speaker, Kuba Gis] matullin, and Nir Lazarovich.

• Thematic program: Fellows of the Institut CNRS Pauli 2015/2016 (fellows 2015/2016)
• Event: Workshop on "Boundaries of groups and representations" (2016)

What is the optimal pinching of curvature on spaces quasiisometric to complex hyperbolic spaces ? This leads to the following problem: what is the best Hölder continuity exponent for a homeomorphism of Euclidean space to a metric space quasisymmetric to the Heisenberg group, when the inverse map is assumed to be Lipschitz ? We give a partial result on this question.

• Thematic program: Fellows of the Institut CNRS Pauli 2015/2016 (fellows 2015/2016)
• Event: Workshop on "Boundaries of groups and representations" (2016)