Related papers: Stationary boundaries on the space of amenable sub…

It is a classical result of Kaimanovich and Vershik and independently of Rosenblatt that a non-amenable group admits a non-degenerate symmetric measure such that the Poisson boundary is trivial. Most if not all examples to date of non-free…

Let $G$ be a locally compact group and $\mu$ an admissible probability measure on $G$. Let $(B,\nu)$ be the universal topological Poisson $\mu$-boundary of $(G,\mu)$ and $\Pi_s(G)$ the universal minimal strongly proximal $G$-flow. This note…

Given a rigid C*-tensor category C with simple unit and a probability measure $\mu$ on the set of isomorphism classes of its simple objects, we define the Poisson boundary of $(C,\mu)$. This is a new C*-tensor category P, generally with…

We show that for any countable group $ G $ equipped with a probability measure $ \mu $, there exists a randomized stopping time $ \tau $ such that $ (G, \mu _{\tau} )$ admits a strictly larger space of bounded harmonic functions than $…

Let $\Gamma$ be a dense countable subgroup of a locally compact continuous group $G$. Take a probability measure $\mu$ on $\Gamma$. There are two natural spaces of harmonic functions: the space of $\mu$-harmonic functions on the countable…

A countable group is C*-simple if its reduced C*-algebra is simple. It is well known that C*-simplicity implies that the amenable radical of the group must be trivial. We show that the converse does not hold by constructing explicit…

In this work we introduce and study a new notion of amenability for actions of locally compact groups on $C^*$-algebras. Our definition extends the definition of amenability for actions of discrete groups due to Claire…

We characterize $C^*$-simplicity for countable groups by means of the following dichotomy. If a group is $C^*$-simple, then the action on the Poisson boundary is essentially free for a generic measure on the group. If a group is not…

Given a minimal action $G\curvearrowright X$ of a countable group $G$ on a compact space $X$, we prove that if the reduced crossed product $G\ltimes_rC(X)$ is simple, then there exists a point whose stabilizer subgroup has trivial amenable…

For each symmetric, aperiodic probability measure $\mu$ on a finitely generated group $G$, we define a subset $A_{\mu}$ consisting of group elements $g$ for which the limit of the ratio ${\mu^{\ast n}(g)}/{\mu^{\ast n}(e)}$ tends to $1$. We…

Given a finitely generated group, the well-known Stability Problem asks whether the non-triviality of the Poisson-Furstenberg boundary (which is equivalent to the existence of non-constant bounded harmonic functions) depends on the choice…

We give sufficient conditions for the non-triviality of the Poisson boundary of random walks on $H(\mathbb{Z})$ and its subgroups. The group $H(\mathbb{Z})$ is the group of piecewise projective homeomorphisms over the integers defined by…

We investigate a class of groups acting on possibly exotic affine buildings $X$ and possessing good proximal properties. Such groups are termed of general type, and their dynamics is analyzed through their flag limit sets in the space of…

We motivate and study the class $\mathcal{C}$ of countable groups $G$ such that the conjugacy relation between minimal actions of $G$ on $\mathbb{R}$ by orientation-preserving homeomorphisms is smooth -- that is, admits a Borel transversal.…

In this paper we study random walks on a finitely generated group $G$ which has a free action on a $\mathbb{Z}^n$-tree. We show that if $G$ is non-abelian and acts minimally, freely and without inversions on a locally finite…

We prove that random walks on Thompson's group $F$ driven by strictly non-degenerate finitely supported probability measures $\mu$ have a non-trivial Poisson boundary. The proof consists in an explicit construction of two different…

We give the first examples of (non-amenable group) amenable actions on stably finite simple C*-algebras. More precisely, we give such actions for any countable group in an explicit way. The main ingredients of our construction are the full…

We study strongly outer actions of discrete groups on C*-algebras in relation to (non)amenability. In contrast to related results for amenable groups, where uniqueness of strongly outer actions on the Jiang-Su algebra is expected, we show…

It is well-known that a Kleinian group is amenable if and only if it is elementary. We establish an analogous property for equivalence relations and foliations with Gromov hyperbolic leaves: they are amenable if and only if they are…

We investigate the conditions under which the space of bounded harmonic functions of a probability measure $\mu$ on a group $G$ is contained in that of another measure $\theta$. We establish that asymptotic commutativity, defined by the…