Related papers: Induced quasi-cocycles on groups with hyperbolical…

Let G be a group admitting a non-elementary acylindrical action on a Gromov hyperbolic space (for example, a non-elementary relatively hyperbolic group, or the mapping class group of a closed hyperbolic surface, or Out(F_n) for n>1). We…

We provide a general sufficient condition for extendability of quasimorphisms on subgroups. This condition recovers the result of Hull--Osin on quasimorphisms on hyperbolically embedded subgroups, and the proof given in this paper is much…

Let $1 \to K \longrightarrow G \stackrel{\pi}\longrightarrow Q$ be an exact sequence of hyperbolic groups. Let $Q_1 < Q$ be a quasiconvex subgroup and let $G_1=\pi^{-1}(Q_1)$. Under relatively mild conditions (e.g. if $K$ is a closed…

$H$ is called a $G$-subgroup of a hyperbolic group $G$ if for any finite subset $M\subset G$ there exists a homomorphism from $G$ onto a non-elementary hyperbolic group $G_1$ that is surjective on $H$ and injective on $M$. In his paper in…

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…

We give technical conditions for a quasi-isometry of pairs to preserve a subgroup being hyperbolically embedded. We consider applications to the quasi-isometry and commensurability invariance of acylindrical hyperbolicity of finitely…

This paper proves under certain conditions the existence of an algorithm, which detects relatively quasiconvex subgroups $H$ of relatively hyperbolic groups $(G,\mathbb{P})$. Additionally, this algorithm outputs an induced peripheral…

For any finitely generated, non-elementary, torsion-free group $G$ that is hyperbolic relative to $\mathbb P$, we show that there exists a group $G^*$ containing $G$ such that $G^*$ is hyperbolic relative to $\mathbb P$ and $G$ is not…

Let G be a group which is hyperbolic relative to a collection of subgroups A, and it is also hyperbolic relative to a collection of subgroups B. Suppose that the collection A contains B. We characterize, for subgroups of G, when…

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 build quasi--isometry invariants of relatively hyperbolic groups which detect the hyperbolic parts of the group; these are variations of the stable dimension constructions previously introduced by the authors. We prove that, given any…

We show that if $G$ is a non-elementary torsion-free word hyperbolic group then there exists another word hyperbolic group $G^*$, such that $G$ is a subgroup of $G^*$ but $G$ is not quasiconvex in $G^*$.

In this paper, we present a notion of quasiconvexity in the setting of finitely-generated groups with hyperbolically embedded subgroups. Our main result shows that this notion yields uniform quasiconvex constants in the setting of coned-off…

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…

Suppose $G$ is a higher-rank connected semisimple Lie group with finite center and without compact factors. Let $\mathbb{G}=G$ or $\mathbb{G}=G\ltimes V$, where $V$ is a finite dimensional vector space $V$. For any unitary representation…

Given a measured space X with commuting actions of two groups G and H satisfying certain conditions, we construct a Hilbert C*(H)-module E(X) equipped with a left action of C*(G), which generalises Rieffel's construction of inducing…

In this m\'emoire we study quasiperiodic cocycles in semi-simple compact Lie groups. For the greatest part of our study, we will focus ourselves to one-frequency cocyles. We will prove that $C^{\infty}$ reducible cocycles are dense in the…

The following discourse is inspired by the works on hyperbolic groups of Epstein, and Neumann/Reeves. Epstein showed that geometrically finite hyperbolic groups are biautomatic. Neumann/Reeves showed that virtually central extensions of…

It is known that every infinite index quasi-convex subgroup $H$ of a non-elementary hyperbolic group $G$ is a free factor in a larger quasi-convex subgroup of $G$. We give a probabilistic generalization of this result. That is, we show that…

Let $G$ be a group hyperbolic relative to a collection of subgroups $\{H_\lambda ,\lambda \in \Lambda \} $. We say that a subgroup $Q\le G$ is hyperbolically embedded into $G$, if $G$ is hyperbolic relative to $\{H_\lambda ,\lambda \in…