Almost finiteness and the small boundary property

TL;DR

This paper establishes a deep connection between the small boundary property and a measure-theoretic form of almost finiteness for free actions of amenable groups, with implications for classifying associated C*-algebras.

Contribution

It introduces the concept of almost finiteness in measure and proves its equivalence to the small boundary property, linking dynamical properties to C*-algebra classification.

Findings

Equivalence of small boundary property and almost finiteness in measure.

Conditions under which almost finiteness, comparison, and m-comparison are equivalent.

Implications for the classification of crossed product C*-algebras and the Toms-Winter conjecture.

Abstract

Working within the framework of free actions of countable amenable groups on compact metrizable spaces, we show that the small boundary property is equivalent to a density version of almost finiteness, which we call almost finiteness in measure, and that under this hypothesis the properties of almost finiteness, comparison, and $m$-comparison for some nonnegative integer $m$ are all equivalent. The proof combines an Ornstein-Weiss tiling argument with the use of zero-dimensional extensions which are measure-isomorphic over singleton fibres. These kinds of extensions are also employed to show that if every free action of a given group on a zero-dimensional space is almost finite then so are all free actions of the group on spaces with finite covering dimension. Combined with recent results of Downarowicz-Zhang and Conley-Jackson-Marks-Seward-Tucker-Drob on dynamical tilings and of Castillejos-Evington-Tikuisis-White-Winter on the Toms-Winter conjecture, this implies that crossed product C$^*$-algebras arising from free minimal actions of groups with local subexponential growth on finite-dimensional spaces are classifiable in the sense of Elliott's program. We show furthermore that, for free actions of countably infinite amenable groups, the small boundary property implies that the crossed product has uniform property $\Gamma$, which under minimality confirms the Toms-Winter conjecture for such crossed products by the aforementioned work of Castillejos-Evington-Tikuisis-White-Winter.