Related papers: Extensions of generic measure-preserving actions

Let $\gp$ be a finite group acting on a compact manifold $M$ and $\maA(M)$ denote the algebra of classical complete symbols on $M$. We determine all traces on the cross-product algebra $\maA(M) \rtimes \Gamma$. These traces appear as…

Let $G$ be a finite group acting transitively on a set $\Omega$. We study what it means for this action to be {\it quasirandom}, thereby generalizing Gowers' study of quasirandomness in groups. We connect this notion of quasirandomness to…

We show that if a countable group $G$ is the free product of infinite abelian groups, then for every free, probability-measure-preserving (p.m.p.) action of $G$, its orbit equivalence class is weakly dense in the space of p.m.p. actions of…

Given a partial action $\alpha=(A_g,\alpha_g)_{g\in \mathcal{G}}$ of a connected groupoid $\mathcal{G}$ on a ring $A$ and an object $x$ of $\mathcal{G}$, the isotropy group $\mathcal{G}(x)$ acts partially on the ideal $A_x$ of $A$ by the…

Del Junco--Lema\'nczyk showed that a generic measure preserving transformation satisfies a certain orthogonality conditions. More precisely, there is a dense $G_\delta$ subset of measure preserving transformations such that for every $T\in…

Each continuous action of a countably infinite discrete group $\Gamma$ on a compact metrizable space X induces a continuous action of $\Gamma$ on the space M(X) of Borel probability measures on X. We compare the local entropy theory for…

We strengthen Marshall Hall's Theorem to show that free groups are locally extended residually alternating. Let F be any free group of rank at least two, let H be a finitely generated subgroup of infinite index in F and let {g_1,...,g_n} be…

We prove that if a countable discrete group $\Gamma$ is {\it w-rigid}, i.e. it contains an infinite normal subgroup $H$ with the relative property (T) (e.g. $\Gamma= SL(2,\Bbb Z) \ltimes \Bbb Z^2$, or $\Gamma = H \times H'$ with $H$ an…

We show sufficient criteria for a group of homeomorphisms acting on a metric space X to extend to one acting on a given compactification of X. We give examples for when this can fail when one of the criteria is not met.

For a metric space $X$ with a compatible measure $\mu$, Genevois and Tessera defined the Scaling Group of $(X,\mu)$ as the subgroup $\Gamma$ of $\mathbb{R}_{>0}$ of positive real numbers $\gamma$ for which there are quasi-isometries of $X$…

Given a variety with a finite group action, we compare its equivariant categorical measure, that is, the categorical measure of the corresponding quotient stack, and the categorical measure of the extended quotient. Using weak factorization…

We introduce a family of atomic measures on free groups generated by no-return random walks. These measures are shown to be very convenient for comparing "relative sizes" of subgroups, context-free and regular subsets (that, subsets…

Motivated by Popa's seminal work \cite{Po04}, in this paper, we provide a fairly large class of examples of group actions $\Gamma \curvearrowright X$ satisfying the extended Neshveyev-St{\o}rmer rigidity phenomenon \cite{NS03}: whenever…

We give an elementary proof of the known fact that every probability measure, defined on an arbitrary $\sigma$-field on a countable sample space $\Omega$, may in fact be extended to a probability measure on the power set of $\Omega$. This…

In the following text we prove that for finite discrete $X$ with at least two elements and infinite $\Gamma$, the generalized shift transformation semigroup $({\mathcal S},X^\Gamma)$ is equicontinuous (resp. has at least an equicontinuous…

We establish multiple recurrence results for pretentious measure-preserving multiplicative actions along generalized Pythagorean triples, that is, solutions to the equation $ax^2 + b y^2 = c z^2$. This confirms the ergodic-theoretic form of…

We introduce infinitesimal weak containment for measure-preserving actions of a countable group $\Gamma$: an action $(X,\mu)$ is infinitesimally contained in $(Y,\nu)$ if the statistics of the action of $\Gamma$ on small measure subsets of…

In this paper we extend the construction of random walks with a prescribed Poisson boundary to the case of measures in the class of a generalized Gibbs state. The price for dropping the $\alpha$-quasiconformal assumptions is that we must…

Given a Woodin cardinal $\delta$, I show that if $F$ is any Easton function with $F"\delta\subseteq\delta$ and $\GCH$ holds, then there is a cofinality-preserving forcing extension in which $2^\gamma= F(\gamma)$ for each regular cardinal…

We consider an ergodic invariant measure $\mu$ for a smooth action of $Z^k$, $k \ge 2$, on a $(k+1)$-dimensional manifold or for a locally free smooth action of $R^k$, $k \ge 2$ on a $(2k+1)$-dimensional manifold. We prove that if $\mu$ is…