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This is an expostion of various aspects of amenability and paradoxical decompositions for groups, group actions and metric spaces. First, we review the formalism of pseudogroups, which is well adapted to stating the alternative of Tarski,…

Consider $ G:= PSL_2(\R)\equiv T^1\H^2$, a modular group $ \Gamma$, and the homogeneous space $ \Gamma\sm G \equiv T^1(\Gamma\sm\H^2)$. Endow $ G $, and then $ \Gamma\sm G $, with a canonical left-invariant metric, thereby equipping it with…

Let G be a locally compact second countable Abelian group. Given a measure preserving action T of G on a standard probability space, let M(T) denote the set of essential values of the spectral multiplicity function of the Koopman unitary…

The classification of compact homogeneous spaces of the form $M=G/K$, where $G$ is a non-simple Lie group, such that the standard metric is Einstein is still open. The only known examples are $4$ infinite families and $3$ isolated spaces…

Our main result is that for all sufficiently large $x_0>0$, the set of commensurability classes of arithmetic hyperbolic 2- or 3-orbifolds with fixed invariant trace field $k$ and systole bounded below by $x_0$ has density one within the…

We consider a non-elementary group action $G \curvearrowright X$ of a locally compact second countable group $G$ on a possibly exotic non-discrete affine building $X$ of type $\tilde{A}_2$. We prove that if $\mu$ is an admissible symmetric…

A first-order theory has the Schroder-Bernstein property if any two of its models that are elementarily bi-embeddable are isomorphic. We prove that if G is an abelian group, then the follwing are equivalent: 1. Th(G, +) has the…

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…

We give a complete characterization of the locally compact groups that are non-elementary Gromov-hyperbolic and amenable. They coincide with the class of mapping tori of discrete or continuous one-parameter groups of compacting…

In ergodic theory, given sufficient conditions on the system, every weak mixing $\mathbb{N}$-action is strong mixing along a density one subset of $\mathbb{N}$. We ask if a similar statement holds in topological dynamics with density one…

This note describes the first example of a group that is amenable, but cannot be obtained by subgroups, quotients, extensions and direct limits from the class of groups locally of subexponential growth. It has a balanced presentation…

We consider the problems of measurable isomorphisms and joinings, measurable centralizers and quotients for certain classes of ergodic group actions on infinite measure spaces. Our main focus is on systems of algebraic origin: actions of…

We prove that, given a torsion-free relatively hyperbolic group G with non-relatively-hyperbolic peripherals, isomorphic finite index subgroups of G have the same index. This applies for instance to fundamental groups of finite-volume…

For \Gamma a countable amenable group consider those actions of \Gamma as measure-preserving transformations of a standard probability space, written as {T_\gamma}_{\gamma \in \Gamma} acting on (X,{\cal F}, \mu). We say…

We consider the field theory of $N$ massless bosons which are free except for an interaction localized on the boundary of their 1+1 dimensional world. The boundary action is the sum of two pieces: a periodic potential and a coupling to a…

We show that for algebraic groups over local fields of characteristic zero, the following are equivalent: Every homomorphism has a closed image, every unitary representation decomposes into a direct sum of finite-dimensional and mixing…

Let Gamma be an S-arithmetic subgroup of a solvable algebraic group G over an algebraic number field F, such that the finite set S contains at least one place that is nonarchimedean. We construct a certain group H, such that if L is any…

We study the automorphism groups of countable homogeneous directed graphs (and some additional homogeneous structures) from the point of view of topological dynamics. We determine precisely which of these automorphism groups are amenable…

A weakly continuous near-action of a Polish group $G$ on a standard Lebesgue measure space $(X,\mu)$ is whirly if for every $A\subseteq X$ of strictly positive measure and every neighbourhood $V$ of identity in $G$ the set $VA$ has full…

Amenability and pseudo-amenability of $ \ell^{1}(S,\omega) $ is characterized, where $S$ is a left (right) zero semigroup or it is a rectangular band semigroup. The equivalence conditions to amenability of $\ell^{1}(S,\omega)$ are provided,…