Related papers: Metric groups, unitary representations and continu…

The unitary representation theory of locally compact contraction groups and their semi-direct products with $\mathbb{Z}$ is studied. We put forward the problem of completely characterising such groups which are type I or CCR and this…

For every $c\geq 1$, we define a strengthening of Kazhdan's Property (T) by considering uniformly bounded representations $\pi$ with fixed bound $|\pi|\leq c$. We carry out a systematic study of this property and show that it can be…

We introduce the notion of uniform exactness, or uniform amenability at infinity, for discrete groups and prove it for a wide class of groups containing free groups and their limit groups. This shows a novel strong convergence phenomenon…

We study unitary representations associated to cocycles of measurable dynamical systems. Our main result establishes conditions on a cocycle, ensuring that ergodicity of the dynamical system under consideration is equivalent to…

By giving an interesting characterisation of amenable multiplicative unitaries in term of one dimensional representations, we show in a simple way that bicrossproducts of amenable locally compact groups is both amenable and coamenable.

We consider model-theoretic properties related to the expressive power of three analogues of $L_{\omega_1, \omega}$ for metric structures. We give an example showing that one of these infinitary logics is strictly more expressive than the…

We first give simplified and corrected accounts of some results in \cite{PiRCP} on compactifications of pseudofinite groups. For instance, we use a classical theorem of Turing \cite{Turing} to give a simplified proof that any definable…

We obtain a complete classification of the continuous unitary representations of the isometry group of the rational Urysohn space $\mathbb{Q}\mathbb{U}$. As a consequence, we show that Isom$(\mathbb{Q}\mathbb{U})$ has property (T). We also…

Consider a proper cocompact CAT(0) space X. We give a complete algebraic characterisation of amenable groups of isometries of X. For amenable discrete subgroups, an even narrower description is derived, implying Q-linearity in the…

This book is concerned with analytic approaches of studying groups and their actions. Much attention is devoted to the study of amenability and Kazhdan's property (T), which are perhaps the most important analytic properties of a group, but…

We prove the existence of a quantum isometry groups for new classes of metric spaces: (i) geodesic metrics for compact connected Riemannian manifolds (possibly with boundary) and (ii) metric spaces admitting a uniformly distributed…

In this paper we study analogues of amenability for topological groups in the context of definable structures. We prove fixed point theorems for such groups. More importantly, we propose definitions for definable actions and continuous…

We study stability of metric approximations of countable groups with respect to groups endowed with ultrametrics, the main case study being a $p$-adic analogue of Ulam stability, where we take $GL_n(\mathbb{Z}_p)$ as approximating groups…

Given an irreducible non-spherical non-affine (possibly non-proper) building $X$, we give sufficient conditions for a group $G < \Aut(X)$ to admit an infinite-dimensional space of non-trivial quasi-morphisms. The result applies to all…

Let $H$ be a proper subgroup of a discrete group $G$. We introduce a notion of relative inner amenability of $H$ in $G$, we prove some equivalent conditions and provide examples as well as counter-examples. We also discuss the corresponding…

We establish a fixed point property for a certain class of locally compact groups, including almost connected Lie groups and compact groups of finite abelian width, which act by simplicial isometries on finite rank buildings with measurable…

We initiate a quantitative study of measure equivalence (and orbit equivalence) between finitely generated groups, which extends the classical setting of $\mathrm L^p$ measure equivalence. In this paper, our main focus will be on amenable…

In this article we analyze the notions of amenability and paradoxical decomposition from an algebraic perspective. We consider this dichotomy for locally finite extended metric spaces and for general algebras over commutative fields. In the…

We study homogeneous metric spaces, by which we mean connected, locally compact metric spaces whose isometry group acts transitively. After a review of some classical results, we use the Gleason-Iwasawa-Montgomery-Yamabe-Zippin structure…

We construct some new cohomology theories for topological groups and Lie groups and study some of its basic properties. For example, we introduce a cohomology theory based on measurable cochains which are continuous in a neighbourhood of…