We say that a group $ G $ is acylindrically hyperbolic if it admits a non-elementary
acylindrical action on a hyperbolic space. We prove that the class of acylindrically hyperbolic …

A discrete group is said to be C*-simple if its reduced C*-algebra is simple, and is said to
have the unique trace property if its reduced C*-algebra has a unique tracial state. A …

A quasi-tree is a geodesic metric space quasi-isometric to a tree. We give a general
construction of many actions of groups on quasi-trees. The groups we can handle include …

We provide new examples of acylindrically hyperbolic groups arising from actions on
simplicial trees. In particular, we consider amalgamated products and HNN-extensions, one …

We define a new notion of contracting element of a group and we show that contracting
elements coincide with hyperbolic elements in relatively hyperbolic groups, pseudo …

We give examples of hyperbolic groups which contain subgroups that are of type $\mathscr
{F} _3 $ but not of type $\mathscr {F} _4 $. These groups are obtained by Dehn filling starting …

We introduce two new tools for studying CAT (0) spaces: curtains, versions of cubical
hyperplanes; and the curtain model, a counterpart of the curve graph. These tools shed new …

Let k be an algebraically closed field. We show that the Cremona group of all birational
transformations of the projective plane $\mathbb {P} _ {\mathbf {k}}^ 2$ is not a simple group …

We consider two manifestations of non-positive curvature: acylindrical actions (on hyperbolic
spaces) and quasigeodesic stability. We study these properties for the class of hierarchically …

This paper is a survey dedicated to the following question: given a group acting on a CAT (0)
cube complex, how to exploit this action to determine whether or not the group is …