Related papers: Equivariant maps to subshifts whose points have sm…

Gao, Jackson, and Seward (see arXiv:1201.0513) proved that every countably infinite group $\Gamma$ admits a nonempty free subshift $X \subseteq \{0,1\}^\Gamma$. Furthermore, a theorem of Seward and Tucker-Drob (see arXiv:1402.4184) implies…

We give an elementary proof for Lewis Bowen's theorem saying that two Bernoulli actions of two free groups, each having arbitrary base probability spaces, are stably orbit equivalent. Our methods also show that for all compact groups K and…

Gao, Jackson, and Seward proved that every countably infinite group $\Gamma$ admits a nonempty free subshift $X \subseteq 2^\Gamma$. Here we strengthen this result by showing that free subshifts can be "large" in various senses.…

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 any infinite countable group $G$ and for any free Borel action $G \curvearrowright X$ there exists an equivariant class-bijective Borel map from $X$ to the free part $\mathrm{Free}(2^G)$ of the $2$-shift $G \curvearrowright…

Ab\'ert-Weiss have shown that the Bernoulli shift s of a countably infinite group \Gamma is weakly contained in any free measure preserving action (mpa) b of \Gamma. We establish a strong version of this result, conjectured by Ioana, by…

We introduce the notion of $(\Gamma,E)$-determinacy for $\Gamma$ a pointclass and $E$ an equivalence relation on a Polish space $X$. A case of particular interest is the case when $E=E_G$ is the (left) shift-action of $G$ on $S^G$ where…

In this paper we study the combinatorics of free Borel actions of the group $\mathbb Z^d$ on Polish spaces. Building upon recent work by Chandgotia and Meyerovitch, we introduce property $F$ on $\mathbb Z^d$-shift spaces $X$ under which…

In this paper, we prove the existence of a particular diagonalization for normal bounded operators defined on subspaces of $L^2(\mathfrak{S})$ where $\mathfrak{S}$ is a second countable LCA group. The subspaces where the operators act are…

We consider the action of a finite subgroup of the mapping class group $Mod(S)$ of an oriented compact surface $S$ of genus $g \geq 2$ on the moduli space $\mathcal{R}(S,G)$ of representations of $\pi_1(S)$ in a connected semisimple real…

Let {a,b} and {c,d} be two pairs of bounding simple closed curves on an oriented surface which intersect nontrivialy. We prove that if these pairs are invariant under the action of an orientation reversing involution, then the corresponding…

We show that, if $\Gamma$ is a point group of $\mathbb{R}^{k+1}$ of order two for some $k\geq 2$ and $\mathcal S$ is a $k$-pseudomanifold which has a free automorphism of order two, then either $\mathcal S$ has a $\Gamma$-symmetric…

The standard double cover of a graph $\Gamma$ is the direct product $\Gamma\times K_2$. A graph $\Gamma$ is said to be stable if all the automorphisms of $\Gamma\times K_2$ come from its factors. Although the study of stability has…

We consider a finite, connected and simple graph $\Gamma$ that admits a vertex-transitive group of automorphisms $G$. Under the assumption that, for all $x \in V(\Gamma)$, the local action $G_x^{\Gamma(x)}$ is the action of…

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…

The symmetrization map $\pi:\mathbb C^2\rightarrow \mathbb C^2$ is defined by $ \pi(z_1,z_2)=(z_1+z_2,z_1z_2). $ The closed symmetrized bidisc $\Gamma$ is the symmetrization of the closed unit bidisc $\overline{\mathbb D^2}$, that is, \[…

We prove that if $G$ is a finitely generated group and $Z$ is a uniformly recurrent subgroup of $G$ then there exists a minimal system $(X,G)$ with $Z$ as its stability system. This answers a query of Glasner and Weiss \cite{GW} in the case…

A countable group \Gamma is called shift-minimal if every non-trivial measure preserving action of \Gamma weakly contained in the Bernoulli shift of \Gamma on ([0,1]^\Gamma ,\lambda ^\Gamma) is free. We show that any group \Gamma whose…

A measure preserving action of a countably infinite group \Gamma is called totally ergodic if every infinite subgroup of \Gamma acts ergodically. For example, all mixing and mildly mixing actions are totally ergodic. This note shows that if…

In this article we study the structure of $\Gamma$-invariant spaces of $L^2(\bf R)$. Here $\bf R$ is a second countable LCA group. The invariance is with respect to the action of $\Gamma$, a non commutative group in the form of a semidirect…