Følner tilings for actions of amenable groups

Abstract

Abstract. We show that every probability-measure-preserving action of a countable amenable group G can be tiled, modulo a null set, using finitely many finite subsets of G ("shapes") with prescribed approximate invariance so that the collection of tiling centers for each shape is Borel. This is a dynamical version of the Downarowicz-HuczekZhang tiling theorem for countable amenable groups and strengthens the Ornstein-Weiss Rokhlin lemma. As an application we prove that, for every countably infinite amenable group G, the crossed product of a generic free minimal action of G on the Cantor set is Z-stable.

Almost Finiteness and the Small Boundary Property

“…Recently Downarowicz and Zhang have verified the hypothesis of Theorem B for groups G whose finitely generated subgroups all have subexponential growth [11]. Conley, Jackson, Marks, Seward, and Tucker-Drob also independently established this fact, in unpublished work, by observing that it follows from a clopen version of the tiling argument in [6] (see the introduction to Section 8). Together with the Z -stability theorem from [25], work of Castillejos-Evington-Tikuisis-White-Winter [4], and the classification results from [12,15,43], this allows us to deduce our main result about C * -crossed products:…”

“…One of the main aims of the present paper is to push this technical connection between measure and topology further at the dynamical level. Our work begins with the observation (Theorem 3.13) that the Ornstein-Weiss tiling argument, as presented in [6], applies equally well to free actions of countable amenable groups on zero-dimensional spaces so as to produce a disjoint collection of towers with clopen levels and Følner shapes such that the part of the space that remains uncovered is small in upper density (or, equivalently, uniformly small on all invariant Borel probability measures). This motivates the concept of almost finiteness in measure for free actions on general compact metrizable spaces, which asks for the same kind of tower decomposition with remainder of small upper density but only requires the levels to be open, as is natural for the purpose of accommodating spaces of higher dimension (Definition 3.5).…”

Almost Finiteness for General Étale Groupoids and Its Applications to Stable Rank of Crossed Products

“…For backgrounds and history of this subject, see the introduction of the paper [23]. Other connections between this subject and C * -algebra theory can be found in [6], [37], [38], [39]. Although the question on genericity no longer makes sense, by adapting Theorem 4.3 of [8] to standard arguments of extensions, one can prove the following statement.…”

“…[6],Theorem 4.2). For any countable amenable group, its generic minimal free actions on the Cantor set admit clopen tower decompositions with arbitrary invariance.…”

A generalized type semigroup and dynamical comparison

“…On the other hand, it is still open whether all continuous actions on the Cantor set of amenable countably infinite groups have comparison. However, by combining Theorem A in [10] and Theorem 4.2 in [5], the property of comparison is generic for minimal free actions of a fixed amenable countably infinite group on the Cantor set. In the setting of non-amenable groups, when there is no invariant measure for the action, the strong boundary actions introduced in [11] and n-filling actions introduced in [14] are natural examples of dynamical comparison.…”