Related papers: Small cancellation in acylindrically hyperbolic gr…

We study the general theory of asymptotically CAT(0) groups, explaining why such a group has finitely many conjugacy classes of finite subgroups, is $F_\infty$ and has solvable word problem. We provide techniques to combine asymptotically…

The automorphisms of a two-generator free group acting on the space of orientation-preserving isometric actions of on hyperbolic 3-space defines a dynamical system. Those actions which preserve a hyperbolic plane but not an orientation on…

In this paper we present a new characterization of free group actions (in classical differential geometry), involving dynamical systems and representations of the corresponding transformation groups. In fact, given a dynamical system, we…

The aim of this note is to give the simplest possible proof that Mapping Class Groups of closed hyperbolic surfaces are acylindrically hyperbolic, and more specifically that their curve graphs are hyperbolic and that pseudo-Anosovs act on…

We establish a criterion that implies the acylindrical hyperbolicity of many Artin groups admitting a visual splitting. This gives a variety of new examples of acylindrically hyperbolic Artin groups, including many Artin groups of FC-type.…

For $\Gamma$ a relatively hyperbolic group, we construct a model for the universal space among $\Gamma$-spaces with isotropy on the family VC of virtually cyclic subgroups of $\Gamma$. We provide a recipe for identifying the maximal…

We construct and classify all groups, given by triangular presentations associated to the smallest thick generalized quadrangle, that act simply transitively on the vertices of hyperbolic triangular buildings of the smallest non-trivial…

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 generalize the graphical small cancellation theory of Gromov to a graphical small cancellation theory over the free product. We extend Gromov's small cancellation theorem to the free product. We explain and generalize Rips-Segev's…

We show that a group that is hyperbolic relative to strongly shortcut groups is itself strongly shortcut, thus obtaining new examples of strongly shortcut groups. The proof relies on a result of independent interest: we show that every…

We show that one can define and effectively compute Stallings graphs for quasi-convex subgroups of automatic groups (\textit{e.g.} hyperbolic groups or right-angled Artin groups). These Stallings graphs are finite labeled graphs, which are…

Shephard groups are common extensions of Artin and Coxeter groups. They appear, for example, in algebraic study of manifolds. An infinite family of Shephard groups which are not Artin or Coxeter groups is considered. Using techniques form…

In the paper `Automorphic functions for a Whitehead-complement group', [Osaka J Math 43 (2006) 63-77] Matsumoto, Nishi and Yoshida constructed automorphic functions on real 3-dimensional hyperbolic space for a Kleinian group called the…

Let C be a one- or two-sided Kazhdan--Lusztig cell in a Coxeter group (W,S), and let Reduced(C) denote the set of reduced expressions of all w in C, regarded as a language over the alphabet S. Casselman has conjectured that Reduced(C) is…

The class of locally compact near abelian groups is introduced and investigated as a class of metabelian groups formalizing and applying the concept of scalar multiplication. The structure of locally compact near abelian groups and its…

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…

In this paper, we study a special class of quasi-homomorphisms, i.e. quasi-retractions from a group to its subgroups. We first give some algebraic and geometric properties of quasi-retracts and then propose a theory of quasi-split short…

In this paper we develop a co-induction operation which transforms an invariant random subgroup of a group into an invariant random subgroup of a larger group. We use this operation to construct new continuum size families of non-atomic,…

Let $A$ be a finite dimensional $Q-$algebra and $\Gamma subset A$ a $Z-$order. We classify those $A$ with the property that $Z^2$ does not embed in $\mathcal{U}(\Gamma)$. We call this last property the hyperbolic property. We apply this in…

The JSJ decomposition encodes the automorphisms and the virtually cyclic splittings of a hyperbolic group. For general finitely presented groups, the JSJ decomposition encodes only their splittings. In this sequence of papers we study the…