We study the geometry of nonrelatively hyperbolic groups. Generalizing a result of Schwartz, any quasi-isometric image of a non-relatively hyperbolic space in a relatively hyperbolic space is contained in a bounded neighborhood of a single peripheral subgroup. This implies that a group being relatively hyperbolic with nonrelatively hyperbolic peripheral subgroups is a quasi-isometry invariant. ...

We propose the metric notion of strong hyperbolicity as a way of obtaining hyperbolicity with sharp additional properties. Specifically, strongly hyperbolic spaces are Gromov hyperbolic spaces that are metrically well-behaved at infinity, and, under weak geodesic assumptions, they are strongly bolic as well. We show that CAT(−1) spaces are strongly hyperbolic. On the way, we determine the best ...

A Kac-Moody algebra is called hyperbolic if it corresponds to a generalized Cartan matrix of hyperbolic type. We study root subsystems of root systems of hyperbolic algebras. In this paper, we classify maximal rank regular hyperbolic subalgebras of hyperbolic Kac-Moody algebras. Introduction A generalized Cartan matrix A is called a matrix of hyperbolic type if it is indecomposable symmetrizabl...

The topological transitivity of non-compact group extensions of topologically mixing subshifts of finite type has been studied recently by Niţică. We build on these methods, and give the first examples of stably transitive non-compact group extensions of hyperbolic dynamical systems. Our examples include extensions of hyperbolic basic sets by the Euclidean group SE(n) for n even, n ≥ 4.

Bowditch showed that a one-ended hyperbolic group which is not a triangle group splits over a two-ended group if and only if its boundary has a local cut point. As a corollary one obtains that splittings of hyperbolic groups over twoended groups are preserved under quasi-isometries. In this note we give a more direct proof of this corollary.

Suppose that all hyperbolic groups are residually finite. The following statements follow: In relatively hyperbolic groups with peripheral structures consisting of finitely generated nilpotent subgroups, quasiconvex subgroups are separable; Geometrically finite subgroups of non-uniform lattices in rank one symmetric spaces are separable; Kleinian groups are subgroup separable. We also show that...

If G is a word hyperbolic group of cohomological dimension 2, then every subgroup of G of type FP2 is also word hyperbolic. Isoperimetric inequalities are denned for groups of type FP2 and it is shown that the linear isoperimetric inequality in this generalized context is equivalent to word hyperbolicity. A sufficient condition for hyperbolicity of a general graph is given along with an applica...

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...

H is called a G-subgroup of a hyperbolic group G if for any finite subset M ⊂ G there exists a homomorphism from G onto a non-elementary hyperbolic group G1 that is surjective on H and injective on M . In his paper in 1993 A. Ol’shanskii gave a description of all G-subgroups in any given non-elementary hyperbolic group G. Here we show that for the same class of G-subgroups the finiteness assump...

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...