There are several ways to generalise the hyperbolic plane and its isometry group to objects in higher dimensions. Perhaps the most familiar is (real) hyperbolic three space, popularised by the work of Thurston [14]. The Poincaré disc and half plane models of the hyperbolic plane naturally come with a complex structure and it is natural to generalise them to complex hyperbolic space in higher co...

The double torus provides a relativistic model for a closed 2D cosmos with topology of genus 2 and constant negative curvature. Its unfolding into an octagon extends to an octagonal tesselation of its universal covering, the hyperbolic space H. The tesselation is analysed with tools from hyperbolic crystallography. Actions on H of groups/subgroups are identified for SU(1, 1), for a hyperbolic C...

We show that every virtually torsion-free subgroup of the outer au-tomorphism group of a conjugacy separable hyperbolic group is residually finite. As a result, we are able to prove that the group of outer automorphisms of every finitely generated Fuchsian group and of every free-by-finite group is residually finite. In an addendum, we also generalise the main result for relatively hyperbolic g...

We call a finitely generated group lacunary hyperbolic if one of its asymptotic cones is an R-tree. We characterize lacunary hyperbolic groups as direct limits of Gromov hyperbolic groups satisfying certain restrictions on the hyperbolicity constants and injectivity radii. Using central extensions of lacunary hyperbolic groups, we solve a problem of Gromov by constructing a group whose asymptot...

For any hyperbolic complex X and a ∈ X we construct a visual metric ď = ďa on ∂X that makes the Isom(X)-action on ∂X bi-Lipschitz, Möbius, symmetric and conformal. We define a stereographic projection of ďa and show that it is a metric conformally equivalent to ďa. We also introduce a notion of hyperbolic dimension for hyperbolic spaces with group actions. Problems related to hyperbolic dimensi...

Let P be a convex polytope in the spherical space S, in the Euclidean space E, or in the hyperbolic space H. Consider the group GP generated by reflections in the facets of P . We call GP a reflection group generated by P . The problem we consider in this paper is to list polytopes generating discrete reflection groups. The answer is known only for some combinatorial types of polytopes. Already...

For every hyperbolic group and more general hyperbolic graphs, we construct an equivariant ideal bicombing: this is a homological analogue of the geodesic flow on negatively curved manifolds. We then construct a cohomological invariant which implies that several Measure Equivalence and Orbit Equivalence rigidity results established in [MSb] hold for all non-elementary hyperbolic groups and thei...

We construct a hyperbolic group with a hyperbolic subgroup for which inclusion does not induce a continuous map of the boundaries. 2010 Mathematics Subject Classification: 20F67

This paper contains examples of closed aspherical manifolds obtained as a by-product of recent work by the author [Bel] on the relative strict hyperbolization of polyhedra. The following is proved. (I) Any closed aspherical triangulated n-manifold M with hyperbolic fundamental group is a retract of a closed aspherical triangulated (n+ 1)manifold N with hyperbolic fundamental group. (II) If B1, ...

We prove the convex combination theorem for hyperbolic n-manifolds. Applications are given both in high dimensions and in 3 dimensions. One consequence is that given two geometrically finite subgroups of a discrete group of isometries of hyperbolic nspace, satisfying a natural condition on their parabolic subgroups, there are finite index subgroups which generate a subgroup that is an amalgamat...