Classifiability of crossed products by nonamenable groups

TL;DR

The paper demonstrates that a broad class of nonamenable groups acting minimally and amenably on compact spaces produce crossed products that are classifiable Kirchberg algebras, extending the understanding of their dynamical and algebraic properties.

Contribution

It establishes dynamical comparison for a wide range of nonamenable groups and shows their crossed products are classifiable Kirchberg algebras in the UCT class.

Findings

All amenable, minimal actions of the considered nonamenable groups have dynamical comparison.

Crossed products are Kirchberg algebras in the UCT class.

These results classify the crossed products via K-theory.

Abstract

We show that all amenable, minimal actions of a large class of nonamenable countable groups on compact metric spaces have dynamical comparison. This class includes all nonamenable hyperbolic groups, many HNN-extensions, nonamenable Baumslag-Solitar groups, a large class of amalgamated free products, lattices in many Lie groups, $\widetilde{A}_2$-groups, as well as direct products of the above with arbitrary countable groups. As a consequence, crossed products by amenable, minimal and topologically free actions of such groups on compact metric spaces are Kirchberg algebras in the UCT class, and are therefore classified by $K$-theory.