Elementary amenability and almost finiteness

TL;DR

The authors prove that every free continuous action of a countably infinite elementary amenable group $G$ on a finite-dimensional compact metrizable space is almost finite, a result with strong regularity consequences: the crossed products for minimal actions are ${\mathcal{Z}}$-stable and classifiable by the Elliott invariant. The core strategy blends a reduction to zero-dimensional dynamics, permanence under finite extensions, and a novel extension-by-$\mathbb{Z}$ argument that fuses Ornstein–Weiss tiling ideas with a careful tower-disjointification process in the semidirect product $H\rtimes\mathbb{Z}$. Two technical lemmas supply precise control over tileability and density needed for the constructions, enabling a recursive, single-scale modification of tower structures. The results significantly extend almost finiteness and ${\mathcal{Z}}$-stability to a broad class of amenable groups and yield classification for the resulting crossed products via the Elliott invariant.

Abstract

We show that every free continuous action of a countably infinite elementary amenable group on a finite-dimensional compact metrizable space is almost finite. As a consequence, the crossed products of minimal such actions are $\mathcal{Z}$-stable and classified by their Elliott invariant.

Theorems & Definitions (15)

• Theorem A
• Corollary B
• Theorem 3.1
• proof
• Lemma 4.1
• proof
• Lemma 4.2
• proof
• Theorem 5.1
• Corollary 5.2