Related papers: Boundaries and equivariant maps for ergodic groupo…

In this paper we develop a theory for Patterson--Sullivan measures for non-Borel Anosov groups on the Furstenberg boundary. Previously, such a theory has been successfully developed for measures supported on the partial flag manifold…

We generalize an equidistribution theorem \`a la Bader-Muchnik for operator-valued measures constructed from a family of boundary representations associated with Gibbs measures in the context of convex cocompact discrete group of isometries…

A result due to M. Gromov states that any two finitely generated groups {\Gamma} and {\Lambda} are quasi-isometric if and only if they admit a topological coupling, i.e., a commuting pair of proper continuous cocompact actions…

We establish convergence in norm and pointwise almost everywhere for the non-conventional (in the sense of Furstenberg) bilinear polynomial ergodic averages \[ A_N(f,g)(x) := \frac{1}{N} \sum_{n =1}^N f(T^nx) g(T^{P(n)}x)\] as $N \to…

The aim of this paper is to prove ergodic decomposition theorems for probability measures quasi-invariant under Borel actions of inductively compact groups (Theorem 1) as well as for sigma-finite invariant measures (Corollary 1). For…

Let $\Gamma$ be a countably infinite group. A common theme in ergodic theory is to start with a probability measure-preserving (p.m.p.) action $\Gamma \curvearrowright (X, \mu)$ and a map $f \in L^1(X, \mu)$, and to compare the global…

In this paper we discuss some connections between measurable dynamics and rigidity aspects of group representations and group actions. A new ergodic feature of familiar group boundaries is introduced, and is used to obtain rigidity results…

We define the Brauer group $\Br(G)$ of a locally compact groupoid $G$ to be the set of Morita equivalence classes of pairs $(\A,\alpha)$ consisting of an elementary C*-bundle $\A$ over $G^{(0)}$ satisfying Fell's condition and an action…

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 introduce computable actions of computable groups and prove the following versions of effective Birkhoff's ergodic theorem. Let $\Gamma$ be a computable amenable group, then there always exists a canonically computable tempered two-sided…

Let $k$ be a local field of characteristic 0, and let $G$ be a connected semisimple almost $k$-algebraic group. Suppose rank$_kG\geq 1$ and $\rho$ is an excellent representation of $G$ on a finite dimensional $k$-vector space $V$. We…

We prove a von Neumann type ergodic theorem for averages of unitary operators arising from the Furstenberg-Poisson boundary representation (the quasi-regular representation) of any lattice in a non-compact connected semisimple Lie group…

Let $G$ be a semi-direct product $G=A\times_\phi K$ with $A$ Abelian and $K$ compact. We characterize spread-out probability measures on $G$ that are mixing by convolutions by means of their Fourier transforms. A key tool is a spectral…

We prove a general result about the decomposition on ergodic components of group actions on boundaries of spherically homogeneous rooted trees. Namely, we identify the space of ergodic components with the boundary of the orbit tree…

A group $\Gamma$ with a family of subgroups $\mathbb{P}$ is relatively hyperbolic if $\Gamma$ admits a cusp-uniform action on a proper $\delta$--hyperbolic space. We show that any two such spaces for a given group pair are quasi-isometric,…

Mimicking the von Neumann version of Kustermans and Vaes' locally compact quantum groups, Franck Lesieur had introduced a notion of measured quantum groupoid, in the setting of von Neumann algebras. In a former article, the author had…

Given a real semisimple connected Lie group $G$ and a discrete subgroup $\Gamma < G$ we prove a precise connection between growth rates of the group $\Gamma$, polyhedral bounds on the joint spectrum of the ring of invariant differential…

Franck Lesieur had introduced in his thesis (now published in an expended and revised version in the {\it M\'emoires de la SMF} (2007)) a notion of measured quantum groupoid, in the setting of von Neumann algebras and a simplification of…

We generalize Bestvina's notion of a $\mathcal{Z}$-boundary for a group to that of a "coarse $\mathcal{Z}$-boundary." We show that established theorems about $\mathcal{Z}$-boundaries carry over nicely to the more general theory, and that…

Given a compact metric space $X$ and a probability measure in the $\sigma-$algebra of Borel subsets of $X$, we will establish a dominated convergence theorem for ultralimits of sequences of integrable maps and apply it to deduce a…