Related papers: Gromov's random monsters do not act non-elementari…

We construct a category of examples of partially hyperbolic geodesic flows which are not Anosov, deforming the metric of a compact locally symmetric space of nonconstant negative curvature. Candidates for such example as the product metric…

This paper proves that in a non-elementary relatively hyperbolic group, the logarithm growth rate of any non-elementary subgroup has a linear lower bound by the logarithm of the size of the corresponding generating set. As a consequence,…

Tree-graded spaces are generalizations of R-trees. They appear as asymptotic cones of groups (when the cones have cut points). Since many questions about endomorphisms and automorphisms of groups, solving equations over groups, studying…

We give upper bounds, linear in rank, to the topological dimensions of the Gromov boundaries of the intersection graph, the free factor graph and the cyclic splitting graph of a finitely generated free group.

We develop methods to control the first-order theory of groups arising as certain direct limits of torsion-free hyperbolic groups, answering several questions in the literature. We construct simple torsion-free Tarski monsters $\Gamma$…

We study the interplay between the algebraic and dynamical properties of groups that admit a general type action on a $\delta$-hyperbolic space such that the induced action on the limit set of the Gromov boundary is faithful. We divide the…

We define hyperbolic groupoids, generalizing the notion of a Gromov hyperbolic group. Examples of hyperbolic groupoids include actions of Gromov hyperbolic groups on their boundaries, pseudogroups generated by expanding self-coverings,…

Let $G$ be a countable group whose action on a metric space $X$ involves a contracting isometry. This setting naturally encompasses groups acting on Gromov hyperbolic spaces, Teichm{\"u}ller space, Culler-Vogtmann Outer space and CAT(0)…

We study discrete, cocompact, isometric actions of groups on Hadamard spaces, and the induced actions on ideal boundaries. For a class of groups generalizing fundamental groups of three-dimensional graph manifolds, we find a set of…

We determine the range of Furstenberg entropy for stationary ergodic actions of nonabelian free groups by an explicit construction involving random walks on random coset spaces.

In this paper, we consider random walks on the isometry groups of general metric spaces. Under some mild conditions, we show that if two non-elementary random walks on a discrete subgroup of the isometry group have non-singular stationary…

We consider discrete random dynamical systems induced by a non-elementary group action on a non-proper hyperbolic space. We prove that if the system is ergodic and satisfies the ``asymptotic past and future independence condition'' as…

A hyperbolic group acts by homeomorphisms on its Gromov boundary. We use a dynamical coding of boundary points to show that such actions are topologically stable in the dynamical sense: any nearby action is semi-conjugate to (and an…

We prove that every non-elementary hyperbolic group $G$ acts with maximal growth on some set $X$ such that every orbit of any element $g \in G$ is finite. As a side-product of our approach we prove that if $G$ is non-elementary hyperbolic,…

In this paper, the first of a series of two, we continue the study of higher index theory for expanders. We prove that if a sequence of graphs is an expander and the girth of the graphs tends to infinity, then the coarse Baum-Connes…

The goal of this article is two-fold: in a first part, we prove Azuma-Hoeffding type concentration inequalities around the drift for the displacement of non-elementary random walks on hyperbolic spaces. For a proper hyperbolic space $M$, we…

The goal of this notice is to establish Not-commutative Point- wise Ergodic Theorems for actions of the Hyperbolic Groups. Similar non-commutative results were done by Bufetov, Khristoforov and Kli- menko, and later by Pollicott and Sharp.…

Our main result is that a finitely generated nilpotent group has no isometric action on an infinite-dimensional Hilbert space with dense orbits. In contrast, we construct such an action with a finitely generated metabelian group.

By a geodesic subspace of a metric space $X$ we mean a subset $A$ of $X$ such that any two points in $A$ can be connected by a geodesic in $A$. It is easy to check that a geodesic metric space $X$ is an $\mathbb{R}$-tree (that is, a…

We investigate the relationship between the metric boundary and the Gromov boundary of a hyperbolic metric space. We show that the Gromov boundary is a quotient topological space of the metric boundary, and that therefore a word-hyperbolic…