Related papers: Combination of quasiconvex subgroups of relatively…

We provide a new method of constructing non-quasiconvex subgroups of hyperbolic groups by utilizing techniques inspired by Stallings' foldings. The hyperbolic groups constructed are in the natural class of right-angled Coxeter groups (RACGs…

Suppose $H$ is a hyperbolic subgroup of a hyperbolic group $G$. Assume there exists $n > 0$ such that the intersection of $n$ essentially distinct conjugates of $H$ is always finite. Further assume $G$ splits over $H$ with hyperbolic vertex…

We prove the Borel Conjecture for a class of groups containing word-hyperbolic groups and groups acting properly, isometrically and cocompactly on a finite dimensional CAT(0)-space.

The paper consists of two parts. In the first one we show that a relatively hyperbolic group $G$ splits as a star graph of groups whose central vertex group is finitely generated and the other vertex groups are maximal parabolic subgroups.…

In this paper we consider convex co-compact subgroups of the projective linear group. We prove that such a group is relatively hyperbolic with respect to a collection of virtually Abelian subgroups of rank two if and only if each open face…

Sela introduced limit groups in his work on the Tarski problem, and showed that each limit group has a cyclic hierarchy. In this paper, a class of relatively hyperbolic groups, equipped with a hierarchy similar to the one for limit groups,…

We prove that the generalised word problem of a finitely generated subgroup of a finitely generated virtually free group is context-free, that a hyperbolic group must be virtually free if it has a torsion-free quasiconvex subgroup of…

A random group contains many quasiconvex surface subgroups.

Let G be a finitely generated relatively hyperbolic group. We show that if no peripheral subgroup of G is hyperbolic relative to a collection of proper subgroups, then the fixed subgroup of every automorphism of G is relatively quasiconvex.…

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)…

We prove the Baum-Connes conjecture for hyperbolic groups and their subgroups.

We prove combination theorems in the spirit of Klein and Maskit in the context of discrete convergence groups acting geometrically finitely on their limit sets. As special cases, we obtain combination theorems for geometrically finite…

We demonstrate the quasi-isometry invariance of two important geometric structures for relatively hyperbolic groups: the coned space and the cusped space. As applications, we produce a JSJ-decomposition for relatively hyperbolic groups…

We prove a combination theorem for hyperbolic groups, in the case of groups acting on complexes displaying combinatorial features reminiscent of non-positive curvature. Such complexes include for instance weakly systolic complexes and…

We develop a theory of convex cocompact subgroups of the mapping class group MCG of a closed, oriented surface S of genus at least 2, in terms of the action on Teichmuller space. Given a subgroup G of MCG defining an extension L_G: 1-->…

We prove that finitely generated relatively hyperbolic groups are bi-exact if and only if all peripheral subgroups are bi-exact. This is a generalization of Ozawa's result which claims that finitely generated relatively hyperbolic groups…

Given a graph of groups $\mathcal{G} = (\Gamma, \{G_v\}, \{G_e\})$ with certain conditions on vertex groups and $G$ acts acylindrically on its Bass-Serre tree $T$. Let $H$ be a finitely generated subgroup of $G$. We prove the following…

The notions of stable and Morse subgroups of finitely generated groups generalize the concept of a quasiconvex subgroup of a word-hyperbolic group. For a word-hyperbolic group $G$, Kapovich provided a partial algorithm which, on input a…

We construct `structure invariants' of a one-ended, finitely presented group that describe the way in which the factors of its JSJ decomposition over two-ended subgroups fit together. For groups satisfying two technical conditions, these…

We show that properly and cocompactly cubulated relatively hyperbolic groups are virtually special, provided the peripheral subgroups are virtually special in a way that is compatible with the cubulation. This extends Agol's result for…