Related papers: Quasiconvexity and relatively hyperbolic groups th…

Given a complex of groups $G(\mathcal{Y}) = (G_\sigma, \psi_a, g_{a,b})$ where all $G_\sigma$ are relatively hyperbolic, the $\psi_a$ are inclusions of full relatively quasiconvex subgroups, and the universal cover $X$ is CAT$(0)$ and…

We study the number of ends of a Schreier graph of a hyperbolic group. Let G be a hyperbolic group and let H be a subgroup of G. In general, there is no algorithm to compute the number of ends of a Schreier graph of the pair (G, H).…

We generalise a theorem of Gersten on surjectivity of the restriction map in $\ell^{\infty}$-cohomology of groups. This leads to applications on subgroups of hyperbolic groups, quasi-isometric distinction of finitely generated groups and…

We prove that any graph of multicurves satisfying certain natural properties is either hyperbolic, relatively hyperbolic, or thick. Further, this geometric characterization is determined by the set of subsurfaces that intersect every vertex…

Given a finitely generated group $G$ that is relatively finitely presented with respect to a collection of peripheral subgroups, we prove that every infinite subgroup $H$ of $G$ that is bounded in the relative Cayley graph of $G$ is…

We consider splittings of groups over finite and two-ended subgroups. We study the combinatorics of such splittings using generalisations of Whitehead graphs. In the case of hyperbolic groups, we relate this to the topology of the boundary.…

Consider a relatively hyperbolic group G. We prove that if G is finitely presented, so are its parabolic subgroups. Moreover, a presentation of the parabolic subgroups can be found algorithmically from a presentation of G, a solution of its…

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…

We consider complex Henon maps which are quasi-hyperbolic. We show that a quasi-hyperbolic map is uniformly hyperbolic if and only if there are no tangencies between stable and unstable manifolds.

This (quasi-)survey addresses the quasi-isometry classification of locally compact groups, with an emphasis on amenable hyperbolic locally compact groups. This encompasses the problem of quasi-isometry classification of homogeneous…

We introduce a new quasi-isometry invariant, called the divergence spectrum, to study finitely generated groups. We compare the concept of divergence spectrum with the other classical notions of divergence and we examine the divergence…

We show that any infinite order element $g$ of a virtually cyclic hyperbolically embedded subgroup of a group $G$ is Morse, that is to say any quasi-geodesic connecting points in the cyclic group $C$ generated by $g$ stays close to $C$.…

We propose the study of Markov chains on groups as a "quasi-isometry invariant" theory that encompasses random walks. In particular, we focus on certain classes of groups acting on hyperbolic spaces including (non-elementary) hyperbolic and…

We study hyperbolicity for quasi-projective varieties where the boundary divisor consists of n+1 numerically parallel effective divisors on a complex projective variety of dimension n, allowing non-empty intersection. Under explicit local…

A simplicial graph is said to be (coarsely) Helly if any collection of pairwise intersecting balls has non-empty (coarse) intersection. (Coarsely) Helly groups are groups acting geometrically on (coarsely) Helly graphs. Our main result is…

Suppose a group $G$ is relatively hyperbolic with respect to a collection $\PP$ of its subgroups and also acts properly, cocompactly on a $\CAT(0)$ (or $\delta$--hyperbolic) space $X$. The relatively hyperbolic structure provides a relative…

Suppose $G$ is a 1-ended finitely generated group that is hyperbolic relative to P a finite collection of 1-ended finitely generated subgroups. Our main theorem states that if the boundary $\partial (G, P)$ has no cut point, then $G$ has…

The asymptotic bound for a length-based attack on the Conjugacy Search Problem in relatively hyperbolic groups is cubic for hyperbolic elements and a "small" polynomial for parabolic elements, depending on the Conjugacy Search Problem for…

Given a domain $G \subsetneq \Rn$ we study the quasihyperbolic and the distance ratio metrics of $G$ and their connection to the corresponding metrics of a subdomain $D \subset G$. In each case, distances in the subdomain are always larger…

We present some results about quasiconvex subgroups of infinite index and their products. After that we extend the standard notion of a subgroup commensurator to an arbitrary subset of a group, and generalize some of the previously known…