Related papers: Rank-one actions, their $(C,F)$-models and constru…

For each {\it well approximable} irrational $\theta$, we provide an explicit rank-one construction of the $e^{2\pi i\theta}$-rotation $R_\theta$ on the circle $\Bbb T$. This solves "almost surely" a problem by del Junco. For {\it every}…

A long-standing open question in the theory of group actions on C*-algebras is the stable rank of the crossed product. Specifically, N. C. Phillips asked that if a finite group $G$ acts on a simple unital C*-algebra $A$ with stable rank…

For topologically mixing locally conformal semigroup actions generated by a finite collection of $C^{1+\alpha}$ conformal local diffeomorphisms, we provide a countable Markov partition satisfying the finite images and the finite cycle…

An isometric compact group action $G \times (M,g) \rightarrow (M,g)$ is called polar if there exists a closed embedded submanifold $\Sigma \subseteq M$ which meets all orbits orthogonally. Let $\Pi$ be the associated generalized Weyl group.…

The unique irreducible representation of $\SL_2(\R)$ on $\R^n$ induces an action, called the \textit{linear action}, of $\SL_2(\Z)$ on the torus $\T^n$ for every $n\geq 2$. For $n$ odd, it factors through $\PSL_2(\Z)$, so we denote by $G_n$…

Suppose $G$ is a connected noncompact locally compact group, $A,B$ are nonempty and compact subsets of $G$, $\mu$ is a left Haar measure on $G$. Assuming that $G$ is unimodular, and $ \mu(A^2) < K \mu(A) $ with $K>1$ a fixed constant, our…

In the paper we discuss the algebraic structure of topological full group $[[T]]$ of a Cantor minimal system $(X,T)$. We show that the topological full group $[[T]]$ has the structure similar to a union of permutational wreath products of…

Let G be a reductive group over a commutative ring R. We say that G has isotropic rank >=n, if every normal semisimple reductive R-subgroup of G contains (G_m)^n. We prove that if G has isotropic rank >=1 and R is a regular domain…

We prove that every acylindrically hyperbolic group admits a minimal and extremely proximal action on a compact metrizable space. If there are no nontrivial finite normal subgroups, then the action is topologically free. This answers…

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,…

For compact Hausdorff admissible right topological (CHART) group $G$, we prove $w(G)=\pi\chi(G)$. This equality is well known for compact topological groups. This implies the criteria for the metrizability of CHART groups: if $G$ is…

By Bekka's theorem the group C*-algebra of an amenable group $G$ is residually finite dimensional (RFD) if and only if $G$ is maximally almost periodic (MAP). We generalize this result in two directions of dynamical flavour. Firstly, we…

In this article, we consider actions of \mathcal{Z}_+^d, \mathcal{R}_+^d and finitely generated free groups on a von Neumann algebras $M$ and prove a version of maximal ergodic inequality. Additionally, we establish non-commutative…

Let $G$ be a group and let ${\mathcal G}$ be a free factor system of $G$, namely a free splitting of $G$ as $G=G_1*\dots*G_k*F_r$. In this paper, we study the set of train track points for ${\mathcal G}$-irreducible automorphisms $\phi$…

Let $G$ be a topological group and let $\mu$ be the Lebesgue measure on the interval $[0,1]$. We let $L_0(G)$ to be the topological group of all $\mu$-equivalence classes of $\mu$-measurable functions defined on [0,1] with values in $G$,…

There is a countable metrizable group acting continuously on the space of rationals in such a way that the only equivariant compactification of the space is a singleton. This is obtained by a recursive application of a construction due to…

The class A of countable groups that admit a faithful, transitive, amenable -- in the sense that there is an invariant mean -- action on a set has been widely investigated in the past. In this paper, we no longer require the action to be…

Given an action of a group G on a topological space X, we establish a necessary and sufficient condition for the existence of a free subgroup F of rank 2 of G acting properly discontinuously on at least one nonempty, open, F-invariant…

We show that, for a given compact or discrete quantum group $G$, the class of actions of $G$ on C*-algebras is first-order axiomatizable in the logic for metric structures. As an application, we extend the notion of Rokhlin property for…

It is proved that for every $r,s \in \mathbb{N}\backslash \{0,1\}$ the action of $\mathbb{F}_{r+s}$ on $\partial_\beta (\mathbb{F}_{r+s}/\mathbb{F}_r)$ is topologically amenable. In particular the $C^*$-algebra associated to the…