Comparison and Almost Finiteness for Actions of Amenable Groups

Abstract

We prove that every action of a countably infinite discrete amenable group on a nonempty compact Hausdorff zero-dimensional space has dynamical comparison, without assuming minimality, freeness, or metrizability. More precisely, strict inequalities under all invariant probability measures imply comparison in the clopen type semigroup with an order-unit remainder. For minimal Cantor actions, the clopen type semigroup is cancellative and almost unperforated, and its canonical map onto the positive cone of the coinvariant group is an isomorphism of ordered monoids. These results answer Melleray's cancellation question and the amenable case of his comparison question. For minimal Cantor actions, we also prove goodness of the associated affine evaluation map and obtain a minimal homeomorphism of the same Cantor space with the same invariant probability measures. We identify the precise obstruction to recovering clopen equidecomposability from affine evaluation alone. This obstruction is given by the infinitesimal subgroup: for every nonempty proper clopen set $A$, it parametrizes the topological-full-group orbits of clopen sets having the same affine evaluation as $A$. Combining comparison with the theorems of Kerr and Szab\'o, we prove that every free action of a countably infinite discrete amenable group on a nonempty compact metrizable space with the topological small boundary property is almost finite. This settles Naryshkin's almost-finiteness conjecture for finite-dimensional compact metrizable spaces. If the action is also minimal, its reduced crossed product is $\mathcal Z$-stable, has nuclear dimension at most one, and satisfies all the regularity conditions in the Toms--Winter conjecture.