# Folner tilings for actions of amenable groups

## TL;DR

This paper proves that actions of countable amenable groups can be finitely tiled with approximate invariance, leading to new results on the structure of crossed products of such actions, including Z-stability.

## Contribution

It extends the tiling theorem to dynamical actions and demonstrates Z-stability of crossed products for generic free minimal actions.

## Key findings

- Every measure-preserving action can be finitely tiled with approximate invariance.
- The tiling centers form Borel sets, ensuring measurability.
- Crossed products of generic free minimal actions are Z-stable.

## 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.