Related papers: A note on complex hyperbolic lattices and strict h…

A class of complex hyperbolic lattices in PU(2,1) called the Deligne-Mostow lattices has been reinterpreted by Hirzebruch and others in terms of line arrangements. They use branched covers over a suitable blow up of the complete…

Let $\Gamma$ be a nonelementary discrete subgroup of SU(n,1) or Sp(n,1). We show that if the trace field of $\Gamma$ is contained in $\mathbb R$, $\Gamma$ preserves a totally geodesic submanifold of constant negative sectional curvature.…

We study deformations of non-cocompact lattices of ${\rm SO}(n,1)$ into ${\rm SU}(n,1)$ and ${\rm SO}(n+1,1)$. A necessary condition for these deformations to remain discrete and faithful (when $n \geqslant 3$) is for the parabolic…

We show that $\Gamma < \textbf{SU}(3,1)$ is a non-elementary complex hyperbolic Kleinian group in which $tr(\gamma) \in \R$ for all $\gamma \in \Gamma$ if and only if $\Gamma$ is conjugate to a subgroup of $\textbf{SO}(3,1)$ or…

Let $G$ be a right-angled Artin group with defining graph $\Gamma$ and let $H$ be a finitely generated group quasi-isometric to $G(\Gamma)$. We show if $G$ satisfies (1) its outer automorphism group is finite; (2) $\Gamma$ does not have…

We introduce a class of countable groups by some abstract group-theoretic conditions. It includes linear groups with finite amenable radical and finitely generated residually finite groups with some non-vanishing $\ell^2$-Betti numbers that…

The hyperbolic space $ \H^d$ can be defined as a pseudo-sphere in the $(d+1)$ Minkowski space-time. In this paper, a Fuchsian group $\Gamma$ is a group of linear isometries of the Minkowski space such that $\H^d/\Gamma$ is a compact…

We prove a theorem describing the limiting fine-scale statistics of orbits of a point in hyperbolic space under the action of a discrete subgroup. Similar results have been proved only in the lattice case, with two recent infinite-volume…

We establish new results on the weak containment of quasi-regular and Koopman representations of a second countable locally compact group $G$ associated with non-singular $G$-spaces. We deduce that any two boundary representations of a…

It is shown that for n bigger than 1, the group of holomorphic isometries of the n dimensional complex hyperbolic space does not admit non-elementary representations into the group of isometries of the infinite dimensional real hyperbolic…

In this paper we continue the analysis of non-diagonalisable hyperbolic systems initiated in \cite{GarJRuz, GarJRuz2}. Here we assume that the system has discontinuous coefficients or more in general distributional coefficients.…

Given a discrete subgroup $\Gamma$ of $PU(1,n)$ it acts by isometries on the unit complex ball $\Bbb{H}^n_{\Bbb{C}}$, in this setting a lot of work has been done in order to understand the action of the group. However when we look at the…

In 1999 Brady constructed the first example of a non-hyperbolic finitely presented subgroup of a hyperbolic group by fibring a non-positively curved cube complex over the circle. We show that his example has Dehn function bounded above by…

We prove that strictly hyperbolized smooth cube manifolds admit normal smooth structures.

A well known question of Gromov asks whether every one-ended hyperbolic group $\Gamma$ has a surface subgroup. We give a positive answer when $\Gamma$ is the fundamental group of a graph of free groups with cyclic edge groups. As a result,…

We show that hyperbolic four-punctured $S^2-$bundles over $S^1$ are distinguished by the finite quotients of their fundamental groups among all 3-manifold groups. To do this, we upgrade a result of Liu to show that the topological type of a…

This paper deals with the $\Gamma$-lattice points problem associated to a discrete subgroup of motions $\Gamma$ in the complex hyperbolic space $\mathbb{C} H^n$. We give two integral formulas for the local average of the number $N(T, z,…

The isometry group of a compact n-dimensional hyperbolic manifold is known to be finite. We show that for every n > 2, every finite group is realized as the full isometry group of some compact hyperbolic n-manifold. The cases n = 2 and n =…

We build examples of properly convex projective manifold $\Omega/ \Gamma$ which have finite volume, are not compact, nor hyperbolic in every dimension $n \geqslant 2$. On the way, we build Zariski-dense discrete subgroups of $\SL_{n+1}(\R)$…

In this article we characterize the complex hyperbolic groups that leave invariant a copy of the Veronese curve in $\Bbb{P}^2_{\Bbb{C}}$. As a corollary we get that every discrete compact surface group in $\PO^+(2,1)$ admits a deformation…