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 groups coincides with many other classes studied in the literature, e.g., the class Cgeom introduced by Hamenstädt, the class of groups admitting a non-elementary weakly properly discontinuous action on a hyperbolic spac...

A geodesic metric space X is called hyperbolic if there exists δ ≥ 0 such that every geodesic triangle ∆ in X is δ-slim, i.e., each side of ∆ is contained in a closed δ-neighborhood of the two other sides. Let G be a group generated by a finite set A and let Γ(G,A) be the corresponding Cayley graph. The group G is said to be word hyperbolic if Γ(G,A) is a hyperbolic metric space. A subset Q of ...

We study automorphisms of a relatively hyperbolic group G. When G is oneended, we describe Out(G) using a preferred JSJ tree over subgroups that are virtually cyclic or parabolic. In particular, when G is toral relatively hyperbolic, Out(G) is virtually built out of mapping class groups and subgroups of GLn(Z) fixing certain basis elements. When more general parabolic groups are allowed, these ...

We prove that in various natural models of a random quotient of a group, depending on a density parameter, for each hyperbolic group there is some critical density under which a random quotient is still hyperbolic with high probability, whereas above this critical value a random quotient is very probably trivial. We give explicit characterizations of these critical densities for the various mod...

Let P be the right-angled hyperbolic dodecahedron or 120-cell, and let W be the group generated by reflections across codimension-one faces of P . We prove that if Γ ⊂ W is a torsion free subgroup of minimal index, then the corresponding hyperbolic manifold Hn/Γ is determined up to homeomorphism by Γ modulo the symmetry group of P .

Open questions 100 1 Appendix. Equivalent definitions of relative hyperbolicity 103 Bibliography 108 2 Chapter 1 Introduction 1.1 Preliminary remarks Originally, the notion of a relatively hyperbolic group was proposed by Gromov [45] in order to generalize various examples of algebraic and geometric nature such as fundamental groups of finite–volume non–compact Riemannian mani-folds of pinched ...

We show that for a hyperbolic knot complement, all but at most 12 Dehn fillings are irreducible with infinite word-hyperbolic fundamental group. AMS Classification numbers Primary: 57M50, 57M27 Secondary: 57M25, 57S25

We study the geometry of non-relatively hyperbolic groups. Generalizing a result of Schwartz, any quasi-isometric image of a non-relatively hyperbolic space in a relatively hyperbolic space is contained in a bounded neighborhood of a single peripheral subgroup. This implies that a group being relatively hyperbolic with non-relatively hyperbolic peripheral subgroups is a quasi-isometry invariant...

We introduce a concept of tree-graded metric space and we use it to show quasi-isometry invariance of certain classes of relatively hyperbolic groups, to obtain a characterization of relatively hyperbolic groups in terms of their asymptotic cones, to find geometric properties of Cayley graphs of relatively hyperbolic groups, and to construct the first example of finitely generated group with a ...

Definitions and classifications of reduced surds for non-arithmetical groups on the hyperbolic plane is proposed, as well those some congruence subgroups. The considered are desymmetrized group, its Γ0 subgroup, a generalized Hecke group according choice commutator subgroup group. definitions studied to ordering generators conjugacy subclasses groups. construction tori defined obtained after te...