Groups with infinite FC-center have the Schmidt property

We show that every countable group with infinite finite conjugacy (FC)-center has the Schmidt property, that is, admits a free, ergodic, measure-preserving action on a standard probability space such that the full group of the associated orbit equivalence relation contains a non-trivial central sequence. As a consequence, every countable, inner amenable group with property (T) has the Schmidt property.

Decompositions and measures on countable Borel equivalence relations

We show that the uniform measure-theoretic ergodic decomposition of a countable Borel equivalence relation $(X, E)$ may be realized as the topological ergodic decomposition of a continuous action of a countable group $\Gamma \curvearrowright X$ generating E. We then apply this to the study of the cardinal algebra $\mathcal {K}(E)$ of equidecomposition types of Borel sets with respect to a compressible countable Borel equivalence relation $(X, E)$ . We also make some general observations regarding quotient topologies on topological ergodic decompositions, with an application to weak equivalence of measure-preserving actions.

Free minimal actions of countable groups with invariant probability measures

We prove that for any countable group $\unicode[STIX]{x1D6E4}$ , there exists a free minimal continuous action $\unicode[STIX]{x1D6FC}:\unicode[STIX]{x1D6E4}\curvearrowright {\mathcal{C}}$ on the Cantor set admitting an invariant Borel probability measure.

Coarse quotients by group actions and the maximal Roe algebra

For a finitely generated group [Formula: see text] acting on a metric space [Formula: see text], Roe defined the warped space [Formula: see text], which one can view as a kind of large scale quotient of [Formula: see text] by the action of [Formula: see text]. In this paper, we generalize this notion to the setting of actions of arbitrary groups on large scale spaces. We then restrict our attention to what we call coarsely discontinuous actions by coarse equivalences and show that for such actions the group [Formula: see text] can be recovered as an appropriately defined automorphism group [Formula: see text] when [Formula: see text] satisfies a large scale connectedness condition. We show that for a coarsely discontinuous action of a countable group [Formula: see text] on a discrete bounded geometry metric space [Formula: see text] there is a relation between the maximal Roe algebras of [Formula: see text] and [Formula: see text], namely that there is a ∗-isomorphism [Formula: see text], where [Formula: see text] is the ideal of compact operators. If [Formula: see text] has Property A and [Formula: see text] is amenable, then [Formula: see text] has Property A, and thus the maximal Roe algebra and full crossed product can be replaced by the usual Roe algebra and reduced crossed product respectively in the above equation.

Recurrence in mean for group actions

In this paper, the mean values of the recurrence are computed for general group actions. Let [Formula: see text] be a metric space with a finite measure [Formula: see text] and [Formula: see text] be a countable group acting on [Formula: see text]. Let [Formula: see text] be a sequence of subsets of [Formula: see text] with [Formula: see text] and put [Formula: see text]. If the Hausdorff measure [Formula: see text] is finite on [Formula: see text] and [Formula: see text] is [Formula: see text]-invariant. We assume that [Formula: see text] and [Formula: see text] are concordant. Then the function [Formula: see text] is [Formula: see text]-integrable and for any [Formula: see text]-measurable set [Formula: see text] we have [Formula: see text] If moreover, [Formula: see text] then [Formula: see text] without the concordance condition for the measure [Formula: see text] and [Formula: see text]

Examples in the entropy theory of countable group actions

Kolmogorov–Sinai entropy is an invariant of measure-preserving actions of the group of integers that is central to classification theory. There are two recently developed invariants, sofic entropy and Rokhlin entropy, that generalize classical entropy to actions of countable groups. These new theories have counterintuitive properties such as factor maps that increase entropy. This survey article focusses on examples, many of which have not appeared before, that highlight the differences and similarities with classical theory.

Random walks on weakly hyperbolic groups

Let G be a countable group which acts by isometries on a separable, but not necessarily proper, Gromov hyperbolic space X. We say the action of G is weakly hyperbolic if G contains two independent hyperbolic isometries. We show that a random walk on such G converges to the Gromov boundary almost surely. We apply the convergence result to show linear progress and linear growth of translation length, without any assumptions on the moments of the random walk. If the action is acylindrical, and the random walk has finite entropy and finite logarithmic moment, we show that the Gromov boundary with the hitting measure is the Poisson boundary.

Diffeomorphism groups of tame Cantor sets and Thompson-like groups

The group of ${\mathcal{C}}^{1}$-diffeomorphisms of any sparse Cantor subset of a manifold is countable and discrete (possibly trivial). Thompson’s groups come out of this construction when we consider central ternary Cantor subsets of an interval. Brin’s higher-dimensional generalizations $nV$ of Thompson’s group $V$ arise when we consider products of central ternary Cantor sets. We derive that the ${\mathcal{C}}^{2}$-smooth mapping class group of a sparse Cantor sphere pair is a discrete countable group and produce this way versions of the braided Thompson groups.

Counting coarse subsets of a countable group

<p>For every countable group G, there are 2^ω distinct classes of coarselyequivalent subsets of G.</p>