Related papers: Hyperbolic groups that are not commensurably coHop…

We prove that affine Coxeter groups, even hyperbolic Coxeter groups and one-ended hyperbolic Coxeter groups are homogeneous in the sense of model theory. More generally, we prove that many (Gromov) hyperbolic groups generated by torsion…

Let $H$ be a torsion-free $\delta$-hyperbolic group with respect to a finite generating set $S$. Let $a_1,..., a_n$ and $a_{1*},..., a_{n*}$ be elements of $H$ such that $a_{i*}$ is conjugate to $a_i$ for each $i=1,..., n$. Then, there is a…

We give a combinatorial criterion that implies both the non-strong relative hyperbolicity and the one-endedness of a finitely generated group. We use this to show that many important classes of groups do not admit a strong relatively…

We describe structure of quasihomomorphisms from arbitrary groups to discrete groups. We show that all quasihomomorphisms are 'constructible', i.e., are obtained via certain natural operations from homomorphisms to some groups and…

We show that if a group is not virtually cyclic and is hyperbolic relative to a family of proper subgroups, then it has a hyperbolically embedded subgroup which contains a finitely generated non-abelian free group as a finite index…

In this article we prove that the set of torsion-free groups acting by isometries on a hyperbolic metric space whose entropy is bounded above and with a compact quotient is finite. The number of such groups can be estimated in terms of the…

In 2002 Polterovich has notably established that on closed aspherical symplectic manifolds, Hamiltonian diffeomorphisms of finite order, which we call Hamiltonian torsion, must in fact be trivial. In this paper we prove the first…

We provide new arguments to see topological Kac-Moody groups as generalized semisimple groups over local fields: they are products of topologically simple groups and their Iwahori subgroups are the normalizers of the pro-p Sylow subgroups.…

We prove that every finitely generated Kleinian group that contains a finite, non-cyclic subgroup either is finite or virtually free or contains a surface subgroup. Hence, every arithmetic Kleinian group contains a surface subgroup.

We prove that if a surface group embeds as a normal subgroup in a K\"ahler group and the conjugation action of the K\"ahler group on the surface group preserves the conjugacy class of a non-trivial element, then the K\"ahler group is…

Let $\Gamma$ be a word hyperbolic group with a cyclic JSJ decomposition that has only rigid vertex groups, which are all fundamental groups of closed surface groups. We show that any group $H$ quasi-isometric to $\Gamma$ is abstractly…

We prove that acylindrically hyperbolic groups are monotileable. That is, every finite subset of the group is contained in a finite tile. This provides many new examples of monotileable groups, and progress on the question of whether every…

The classes of abelian groups that are (uniformly) strongly Hopfian abelian groups, and dually, (uniformly) strongly co-Hopfian abelian groups have been studied by several authors, including Abdelalim (2015) and Abdelalim-Chillali-Essanouni…

This paper is devoted to the first-order theory of torsion-free hyperbolic groups. One of its purposes is to review some results and to provide precise and correct statements and definitions, as well as some proofs and new results. A key…

We prove that for a one-ended hyperbolic graph $X$, the size of the quotient $X/G$ by a group $G$ acting freely and cocompactly bounds from below the number of simplices in an Eilenberg-MacLane space for $G$. We apply this theorem to show…

We give a complete characterization of torsion-free hyperbolic groups which are homogeneous in the sense of first-order logic, in terms of the JSJ decompositions of their free factors.

We give a complete characterization of the locally compact groups that are non-elementary Gromov-hyperbolic and amenable. They coincide with the class of mapping tori of discrete or continuous one-parameter groups of compacting…

A simple surface amalgam is the union of a finite collection of surfaces with precisely one boundary component each and which have their boundary curves identified. We prove if two fundamental groups of simple surface amalgams act properly…

Relying on the techniques and ideas from our recent paper [13], we prove several anti-classification results for various rigidity conditions in countable abelian and nilpotent groups. We prove three main theorems: (1) the rigid abelian…

We show that there exist infinitely many commensurability classes of finite volume hyperbolic 3-manifolds whose fundamental group contains a subgroup which is locally free but not free. The main technical tool is the fact that a collection…