Folner tilings for actions of amenable groups

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--Huczek--Zhang 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.

Minor revisions. Final version

References in corpus (4)

• Quasidiagonality of nuclear C*-algebras
• Decomposition rank of UHF-absorbing C*-algebras
• The Rokhlin dimension of topological Z^m-actions
• Genericity in Topological Dynamics

Cited by in corpus (9)

• Nuclear dimension of simple C*-algebras
• Almost finiteness and the small boundary property
• Nuclear dimension of simple stably projectionless C*-algebras
• Almost finiteness for general etale groupoids and its applications to stable rank of crossed products
• The comparison property of amenable groups
• Amenable wreath products with non almost finite actions of mean dimension zero
• Classification of tiling $C^{\ast }$-algebras
• Kolmogorov complexity and entropy of amenable group actions
• Elementary amenability and almost finiteness