Essential freeness, allostery and $\mathcal{Z}$-stability of crossed products

Abstract

We explore classifiability of crossed products of actions of countable amenable groups on compact, metrizable spaces. It is completely understood when such crossed products are simple, separable, unital, nuclear and satisfy the UCT: these properties are equivalent to the combination of minimality and topological freeness, and the challenge in this context is establishing $\mathcal{Z}$-stability. While most of the existing results in this direction assume freeness of the action, there exist numerous natural examples of minimal, topologically free (but not free) actions whose crossed products are classifiable. In this work, we take the first steps towards a systematic study of $\mathcal{Z}$-stability for crossed products beyond the free case, extending the available machinery around the small boundary property and almost finiteness to a more general setting. Among others, for actions of groups of polynomial growth with the small boundary property, we show that minimality and topological freeness are not just necessary, but also \emph{sufficient} conditions for classifiability of the crossed product. Our most general results apply to actions that are essentially free, a property weaker than freeness but stronger than topological freeness in the minimal setting. Very recently, M. Joseph produced the first examples of minimal actions of amenable groups which are topologically free and not essentially free. While the current machinery does not give any information for his examples, we develop ad-hoc methods to show that his actions have classifiable crossed products.