Almost 1-1 extensions of Furstenberg-Weiss type and test for amenability

Abstract

Let $G$ be an infinite residually finite group. We show that for every minimal equicontinuous Cantor system $(Z,G)$ with a free orbit, and for every minimal extension $(Y,G)$ of $(Z,G)$, there exist a minimal almost 1-1 extension $(X,G)$ of $(Z,G)$ and a Borel equivariant map $\psi :Y\to X$ that induces an affine bijection ${\psi }^{\ast }$ between $M(Y,G)$ and $M(X,G)$, the spaces of invariant probability measures of $(Y,G)$ and $(X,G)$, respectively. If $Y$ is a Cantor set, then $(Y,G)$ and $(X,G)$ are Borel isomorphic, i.e., ${\psi }^{\ast }$ is also a homeomorphism. As an application, we show that the family of Toeplitz subshifts is a test for amenability for residually finite groups, i.e., a residually finite group $G$ is amenable if and only if every Toeplitz $G$-subshift has invariant probability measures.

arXiv comment19 pages. arXiv admin note: substantial text overlap with arXiv:2312.12562, We correct an error in the proof of Theorem 1. We have changed the title and the introduction, in order to emphasis the existence of Furstenberg-Weiss type almost 1-1 extensions that preserve measures