Polynomial growth, comparison, and the small boundary 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 $C^{\ast }$-crossed product is $\mathcal{Z}$-stable, and consequently that such crossed products are classified by their Elliott invariant.

arXiv comment2nd version: fixed mistakes in the proofs; added Lemma 3.1. 3rd version: added an acknowledgement