Related papers: Relatively hyperbolic groups with planar boundarie…

A cocompact lattice in a semisimple Lie group $G$ is a discrete subgroup $\Gamma$ such that the quotient $G/\Gamma$ is compact. Does such a lattice always contain a surface group, i.e. a subgroup isomorphic to the fundamental group of a…

We explore the combination theorem for a group G splitting as a graph of relatively hyperbolic groups. Using the fine graph approach to relative hyperbolicity, we find short proofs of the relative hyperbolicity of G under certain…

Let $G$ be a non-elementary word-hyperbolic group acting as a convergence group on a compact metrizable space $Z$ so that there exists a continuous $G$-equivariant map $i:\partial G\to Z$, which we call a \emph{Cannon-Thurston map}. We…

Let S be a closed surface of genus at least 2. We show that a finitely generated group G which is an extension of the fundamental group H of S is word hyperbolic if and only the orbit map of the quotient group G/H on the complex of curves…

This paper proves the sandwich classification conjecture for subgroups of an even dimensional hyperbolic unitary group $U_{2n}(R,\Lambda)$ which are normalized by the elementary subgroup $EU_{2n}(R,\Lambda)$, under the condition that $R$ is…

We study the action of a relatively hyperbolic group on its boundary, by methods of symbolic dynamics. Under a condition on the parabolic subgroups, we show that this dynamical system is finitely presented. We give examples where this…

Let $Sp(2,1)$ be the isometry group of the quaternionic hyperbolic plane ${{\bf H}_{\mathbb H}}^2$. An element $g$ in $Sp(2,1)$ is `hyperbolic' if it fixes exactly two points on the boundary of ${{\bf H}_{\mathbb H}}^2$. We classify pairs…

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…

Take two isomorphic convex co-compact co-infinite volume Kleinian groups, whose regular sets are diffeomorphic. The quotient of hyperbolic 3-space by these groups gives two hyperbolic 3-manifolds whose scattering operators may be compared.…

We construct an uncountable sequence of groups acting uniformly properly on hyperbolic spaces. We show that only countably many of these groups can be virtually torsion-free. This gives new examples of groups acting uniformly properly on…

The aim of this short note is to provide a proof of the decidability of the generalized membership problem for relatively quasi-convex subgroups of finitely presented relatively hyperbolic groups, under some reasonably mild conditions on…

We study different notions of quasiconvexity for a subgroup $H$ of a relatively hyperbolic group $G.$ The first result establishes equivalent conditions for $H$ to be relatively quasiconvex. As a corollary we obtain that the relative…

Given a finitely presented group $G$ and a surjective homomorphism $G\to \mathbb{Z}^n$ with finitely presented kernel $K$, we give an upper bound on the Dehn function of $K$ in terms of an area-radius pair for $G$. As a consequence we…

We develop the theory of Patterson-Sullivan measures on the boundary of a locally compact hyperbolic group, associating to certain left invariant metrics on the group measures on the boundary. We later prove that for second countable,…

Generalizing a classical theorem of Carlson and Toledo, we prove that any Zariski dense isometric action of a K\"{a}hler group on the real hyperbolic space of dimension at least 3 factors through a homomorphism onto a cocompact discrete…

Let Gamma be a torsion-free group which is hyperbolic relative to a collection of free abelian subgroups. We construct Makanin-Razborov diagrams for Gamma. We also prove that every system of equations over Gamma is equivalent to a finite…

We study relations between maps between relatively hyperbolic groups/spaces and quasisymmetric embeddings between their boundaries. More specifically, we establish a correspondence between (not necessarily coarsely surjective)…

For a hyperbolic subgroup H of a hyperbolic group G, we describe sufficient criteria to guarantee the following. 1) Geodesic rays in H starting at the identity land at a unique point of the boundary of G. 2)The inclusion of H into G does…

In this paper, we prove a combination theorem for a relatively acylindrical graph of relatively hyperbolic groups (Theorem 1.1). Here, we are extending the technique of [Tom21] and constructing Bowditch boundary of the fundamental group of…

We study those groups that act properly discontinuously, cocompactly, and isometrically on CAT(0) spaces with isolated flats and the Relative Fellow Traveller Property. The groups in question include word hyperbolic CAT(0) groups as well as…