Related papers: Boundary dynamics, triple transitivity, and mixed …

We prove that random groups in the Gromov density model, at any density, satisfy property (FA), i.e. they do not act non-trivially on trees. This implies that their Gromov boundaries, defined at density less than 1/2, are Menger curves.

Let K be a fine hyperbolic graph and G be a group acting on K with finite quotient. We prove that G is exact provided that all vertex stabilizers are exact. In particular, a relatively hyperbolic group is exact if all its peripheral groups…

Using ultrapowers of $C^{\ast}$-algebras we provide a new construction of the multiplier algebra of a $C^{\ast}$-algebra. This extends the work of Avsec and Goldbring [Houston J. Math., to appear, arXiv:1610.09276.] to the setting of…

Stable subgroups and the Morse boundary are two systematic approaches to collect and study the hyperbolic aspects of finitely generated groups. In this paper we unify and generalize these strategies by viewing any geodesic metric space as a…

This is the second of two papers but has been written so as to have minimal dependence on the first paper (which is also on this archive). Let G be a group and let M be a CAT(0) proper metric space (e.g. a simply connected complete…

A measure preserving action of a countably infinite group \Gamma is called totally ergodic if every infinite subgroup of \Gamma acts ergodically. For example, all mixing and mildly mixing actions are totally ergodic. This note shows that if…

The algebras of the symmetry operators for the Hamilton-Jacobi and Klein-Gordon-Fock equations are found for a charged test particle moving in an external electromagnetic field in a spacetime manifold, on the isotropic (null) hypersurface…

We show that the Gromov boundary of the free product of two infinite hyperbolic groups is uniquely determined up to homeomorphism by the homeomorphism types of the boundaries of its factors. We generalize this result to graphs of hyperbolic…

We prove that any action of a higher rank lattice on a Gromov-hyperbolic space is elementary. More precisely, it is either elliptic or parabolic. This is a large generalization of the fact that any action of a higher rank lattice on a tree…

We prove that the Poisson boundary of a random walk with finite entropy on a non-elementary hyperbolic group can be identified with its hyperbolic boundary, without assuming any moment condition on the measure. We also extend our method to…

Let $X$ be a proper geodesic metric space and let $G$ be a group of isometries of $X$ which acts geometrically. Cordes constructed the Morse boundary of $X$ which generalizes the contracting boundary for CAT(0) spaces and the visual…

Let $\phi$ be an atoroidal outer automorphism of the free group $F_n$. We study the Gromov boundary of the hyperbolic group $G_{\phi} = F_n \rtimes_{\phi} \mathbb{Z}$. We explicitly describe a family of embeddings of the complete bipartite…

We study profinite actions of residually finite groups in terms of weak containment. We show that two strongly ergodic profinite actions of a group are weakly equivalent if and only if they are isomorphic. This allows us to construct…

We introduce and investigate several quantum group dynamical notions for the purpose of studying $C^*$-simplicity of discrete quantum groups via the theory of boundary actions. In particular we define a quantum analogue of Powers' Averaging…

We provide a unifying approach which links results on algebraic actions by Lind and Schmidt, Chung and Li, and a topological result by Meyerovitch that relates entropy to the set of asymptotic pairs. In order to do this we introduce a…

With the aid of a Gromov hyperbolic characterization of uniform domains, we first give an affirmative answer to an open question arisen by V\"ais\"al\"a under weaker assumption. Next, we show that the three-point condition introduced by…

Let $G$ be a countable discrete amenable group, ${\cal M}$ a McDuff factor von Neumann algebra, and $A$ a separable nuclear weakly dense C$^*$-subalgebra of ${\cal M}$. We show that if two centrally free actions of $G$ on ${\cal M}$ differ…

The main aim of the present set of notes is to give new, short and essentially self-contained proofs of some classical, as well as more recent, results about random walks on groups. For instance, we shall see that the drift characterization…

We relate the topology of the Morse boundary of a group to geometric and algorithmic properties of the group. In particular, we show that a group has $\sigma$-compact Morse boundary if and only if it is Morse local-to-global. We also…

We study the action of a relatively hyperbolic group on its boundary, by methods of symbolic dynamics. Under a condition on the parabolic subgroups, we show that this dynamical system is finitely presented. We give examples where this…