Polynomial growth, comparison, and the small boundary property

TL;DR

This paper proves that minimal actions of polynomial growth groups on compact spaces have the comparison property, leading to classification results for their associated $C^*$-algebras under certain conditions.

Contribution

It establishes the comparison property for minimal polynomial growth group actions and links the small boundary property to $ ext{Z}$-stability and classification of crossed products.

Findings

Actions with small boundary property are almost finite.

Crossed products are $ ext{Z}$-stable and classifiable by Elliott invariants.

Polynomial growth groups induce actions with the comparison property.

Abstract

We show that a minimal action of a finitely generated group of polynomial growth on a compact metrizable space has comparison. It follows that if such an action has the small boundary property then it is almost finite and its $C^*$-crossed product is $\mathcal{Z}$-stable, and consequently that such crossed products are classified by their Elliott invariant.