Related papers: The verbal width of acylindrically hyperbolic grou…

Formal languages based on the multiplication tables of finitely generated groups are investigated and used to give a linguistic characterization of word hyperbolic groups.

We show that Morse elements are generic in acylindrically hyperbolic groups. As an application, we observe that fully irreducible outer automorphisms are generic in the outer automorphism group of a finite-rank free group.

It is shown that there exist finitely generated infinite simple groups of infinite commutator width and infinite square width on which there exists no stably unbounded conjugation-invariant norm, and in particular stable commutator length…

A subset of a group is characteristic if it is invariant under every automorphism of the group. We study word length in fundamental groups of closed hyperbolic surfaces with respect to characteristic generating sets consisting of a finite…

We give examples of hyperbolic groups with finite-rank free subgroups of huge (Ackermannian) distortion.

We find strictly ascending HNN extensions of finite rank free groups possessing a presentation 2-complex which is a non positively curved square complex. On showing these groups are word hyperbolic, we have by results of Wise and Agol that…

We prove that there are finite area flat surfaces whose Veech group is an infinite cyclic group consisting of hyperbolic elements

We prove that, for a finitely generated group hyperbolic relative to virtually abelian subgroups, the generalised word problem for a parabolic subgroup is the language of a real-time Turing machine. Then, for a hyperbolic group, we show…

For any finite collection $f_i$ of fully irreducible automorphisms of the free group $F_n$ we construct a connected $\delta$-hyperbolic $Out(F_n)$-complex in which each $f_i$ has positive translation length.

We prove that every finitely presented group with positive first $\ell^2$-Betti number that virtually surjects onto $\mathbb Z$ is acylindrically hyperbolic. In particular, this implies acylindrical hyperbolicity of finitely presented…

For a fixed word hyperbolic group we compare different residual properties related to quasiconvex subgroups.

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 prove that for a finitely generated subgroup $H$ of a word-hyperbolic group $G$ the Frattini subgroup $F(H)$ of $H$ is finite.

We give a dynamical characterization of acylindrically hyperbolic groups. As an application, we prove that non-elementary convergence groups are acylindrically hyperbolic.

We prove that the Fesenko Groups, which occur as natural closed subgroups of the Nottingham Group, have finite width and infinite obliquity.

In this paper, we study the $C$-width of HNN extension of a group via its proper isomorphic subgroups and amalgamated free product of two groups via their proper isomorphic subgroups with respect to conjugation invariant generating set. We…

The goal of this article is to survey some recent developments in the study of groups acting on hyperbolic spaces. We focus on the class of acylindrically hyperbolic groups; it is broad enough to include many examples of interest, yet a…

Recently George Bergman proved that the symmetric group of an infinite set possesses the following property which we call by the {\it universality of finite width}: given any generating set $X$ of the symmetric group of an infinite set…

We prove that every countable family of countable acylindrically hyperbolic groups has a common finitely generated acylindrically hyperbolic quotient. As an application, we obtain an acylindrically hyperbolic group $Q$ with strong fixed…

If there is a non-residually finite hyperbolic group, then there is a non-residually finite rigid hyperbolic group.