Related papers: Topological and geometric restrictions on hypercon…

Given a finite graph of relatively hyperbolic groups with its fundamental group relatively hyperbolic and edge groups quasi-isometrically embedded and relatively quasiconvex in vertex groups, we prove that vertex groups are relatively…

We show that a Kleinian surface group, or hyperbolic 3-manifold with a cusp-preserving homotopy-equivalence to a surface, has bounded geometry if and only if there is an upper bound on an associated collection of coefficients that depend…

We study geometry, topology and deformation spaces of noncompact complex hyperbolic manifolds (geometrically finite, with variable negative curvature), whose properties make them surprisingly different from real hyperbolic manifolds with…

We define a new family of discrete representations of relatively hyperbolic groups which unifies many existing definitions and examples of geometrically finite behavior in higher rank. The definition includes the relative Anosov…

In this paper, we study the geometric and dynamical properties of maximal representations of surface groups into Hermitian Lie groups of rank 2. Combining tools from Higgs bundle theory, the theory of Anosov representations, and…

A vanishing theorem for a convex cocompact hyperbolic manifold is established, which relates the L2 cohomology to the Hausdorff dimension of the limit set. The borderline case is shown to characterize the manifold completely.

Carrier graphs of groups representing subgroups of a given relatively hyperbolic groups are introduced and a combination theorem for relatively quasi-convex subgroups is proven. Subsequently a theory of folds for such carrier graphs is…

For every dimension d, there is an infinite family of convex co-compact reflection groups of isometries of hyperbolic d-space --- the superideal (simplicial and cubical) reflection groups --- with the property that a random group at any…

We interpret the Hilbert entropy of a convex projective structure on a closed higher-genus surface as the Hausdorff dimension of the non-differentiability points of the limit set in the full flag space $\mathcal F(\mathbb R^3)$.…

We characterize groups admitting Anosov representations into $\mathsf{SL}(3,\mathbb R)$, projective Anosov representations into $\mathsf{SL}(4,\mathbb R)$, and Borel Anosov representations into $\mathsf{SL}(4,\mathbb R)$. More generally, we…

Wigner's unitary representation of the Lorentz group is extended to a representation of the complex orthosymplectic Lie super group OSp_C(1|2) acting on Minkowski (3,1|4)-dimensional super space essentially by Hermitean conjugation. The…

We demonstrate the quasi-isometry invariance of two important geometric structures for relatively hyperbolic groups: the coned space and the cusped space. As applications, we produce a JSJ-decomposition for relatively hyperbolic groups…

In this paper we show that bending a finite volume hyperbolic $d$-manifold $M$ along a totally geodesic hypersurface $\Sigma$ results in a properly convex projective structure on $M$ with finite volume. We also discuss various geometric…

These notes are an extended version of a talk given by the author in the seminar "Theorie Spectrale et Geometrie" at the Institut Fourier in No- vember 2016. We present here some aspects of a work in collaboration with B. Collier and N.…

In this paper, we investigate the Tomas-Stein restriction estimates on convex cocompact hyperbolic manifolds $\Gamma\backslash\mathbb{H}^{n+1}$. Via the spectral measure of the Laplacian, we prove that the Tomas-Stein restriction estimate…

Considering an integer $d>0$, we show the existence of convex-cocompactrepresentations of surface groups into SO(4,1) admitting an embedded minimal map withcurvatures in $(-1,1)$ and whose associated hyperbolic 4-manifolds are disk bundles…

For n>3 we study spaces obtained from finite volume complete real hyperbolic n-manifolds by removing a compact totally geodesic submanifold of codimension two. We prove that their fundamental groups are relative hyperbolic, co-Hopf,…

Let X be an arbitrary hyperbolic geodesic metric space and let G be a countable non-elementary weakly acylindrical group of isometries of X. We show that the second bounded cohomology group of G with real coefficients or with coefficients…

We prove uniform boundedness of certain boundary representations on appropriate fractional Sobolev spaces $W^{s,p}$ with $p>1$ for arbitrary Gromov hyperbolic groups. These are closed subspaces of $L^p$ and in particular Hilbert spaces in…

We study convex sets C of finite (but non-zero volume in Hn and En. We show that the intersection of any such set with the ideal boundary of Hn has Minkowski (and thus Hausdorff) dimension of at most (n-1)/2, and this bound is sharp. In the…