Related papers: Lacunary sets for actions of tsi groups

If $\Gamma$ is a discrete subgroup of $PSL(3,\Bbb{C})$, it is determined the equicontinuity region $Eq(\Gamma)$ of the natural action of $\Gamma$ on $\Bbb{P}^2_\Bbb{C}$. It is also proved that the action restricted to $Eq(\Gamma)$ is…

In this paper we prove that if there is a Borel-cover gamma-set of cardinality the continuum, then there is one which is not hereditary. A set of reals X is a Borel-cover gamma-set iff for every countable family of Borel sets which is an…

Let G=SL(n,R) with n>5. We construct examples of lattices Gamma of G, subgroup A of the diagonal group and points x in G/Gamma such that the closure of the orbit Ax is not homogeneous but does not factors through the action of a…

Let $\Gamma$ be a countably infinite group. Given $k \in \mathbb{N}$, we use $\mathrm{Free}(k^\Gamma)$ to denote the free part of the Bernoulli shift action of $\Gamma$ on $k^\Gamma$. Seward and Tucker-Drob showed that there exists a free…

We show that every non-precompact topological group admits a fixed point-free continuous action by affine isometries on a suitable Banach space. Thus, precompact groups are defined by the fixed point property for affine isometric actions on…

A topological space is defined to be banalytic (resp. analytic) if it is the image of a Polish space under a Borel (resp. continuous) map. A regular topological space is analytic if and only if it is banalytic and cosmic. Each (regular)…

Given a conjugacy class $\mathcal{C}$ in a group $G$ we define a new graph, $\Gamma(\mathcal{C})$, whose vertices are elements of $\mathcal{C}$; two vertices $g,h\in \mathcal{C}$ are connected in $\Gamma(\mathcal{C})$ if $[g,h]=1$ and…

Working in the framework of Borel reducibility, we study various notions of embeddability between groups. We prove that the embeddability between countable groups, the topological embeddability between (discrete) Polish groups, and the…

Let $\Gamma$ be a countably infinite discrete group. A $\Gamma$-flow $X$ (i.e., a nonempty compact Hausdorff space equipped with a continuous action of $\Gamma$) is called $S$-minimal for a subset $S \subseteq \Gamma$ if the partial orbit…

We show that for a $\sigma $-ideal $\ci$ with a Borel base of subsets of an uncountable Polish space, if $\ca$ is (in several senses) a "regular" family of subsets from $\ci $ then there is a subfamily of $\ca$ whose union is completely…

We present a simple approach to questions of topological orbit equivalence for actions of countable groups on topological and smooth manifolds. For example, for any action of a countable group $\Gamma$ on a topological manifold where the…

An \textit{algebraic} action of a discrete group $\Gamma $ is a homomorphism from $\Gamma $ to the group of continuous automorphisms of a compact abelian group $X$. By duality, such an action of $\Gamma $ is determined by a module…

We study the Borel subsets of the plane that can be made closed by refining the Polish topology on the real line. These sets are called potentially closed. We first compare Borel subsets of the plane using products of continuous functions.…

We show that for any Polish group $G$ and any countable normal subgroup $\Gamma\triangleleft G$, the coset equivalence relation $G/\Gamma$ is a hyperfinite Borel equivalence relation. In particular, the outer automorphism group of any…

Let $\Gamma$ be a sofic group with a copy of $\mathbb{Z}$ in its center. We construct an uncountable family of pairwise nonisomorphic measure-preserving $\Gamma$ actions with completely positive entropy, none of which is a factor of a…

We study when a continuous isometric action of a Polish group on a complete metric space is, or can be, transitive. Our main results consist of showing that certain Polish groups, namely $\mathrm{Aut}^*(\mu)$ and $\mathrm{Homeo}^+[0,1]$,…

We show that the $\mathrm{Hom}$ functor from the category $\mathbf{LCPAb}$ of locally compact Polish abelian groups to the category $\mathbf{PAb}$ of Polish abelian groups has a total right derived functor, improving on Hoffmann and…

Let~$\mathcal{A}$ be an almost disjoint family of subsets of an infinite set~$\Gamma$, and denote by~$X_{\mathcal{A}}$ the closed subspace of~$\ell_\infty(\Gamma)$ spanned by the indicator functions of intersections of finitely many sets…

Let xi be a non-null countable ordinal. We study the Borel subsets of the plane that can be made $\bormxi$ by refining the Polish topology on the real line. These sets are called potentially $\bormxi$. We give a Hurewicz-like test to…

Let $\Gamma$ be a finitely generated group which admits an action by homeomorphisms on a compact metrizable space $X$. We show that there is a metric on $X$ defining the original topology such that for this metric, the action is by…