Compact actions whose orbit equivalence relations are not profinite

TL;DR

This paper demonstrates that certain non-profinite group actions with spectral gap cannot be simplified to modular actions, providing new examples of complex orbit structures in compact group actions.

Contribution

It establishes the non-existence of Borel homomorphisms from specific non-profinite actions with spectral gap to modular actions, and constructs new examples of compact actions that are antimodular and not orbit equivalent to profinite actions.

Findings

Non-profinite actions with spectral gap are not orbit equivalent to modular actions.

Constructs the first examples of compact actions that are antimodular.

Answers open questions by Kechris and Tsankov regarding orbit equivalence.

Abstract

Let $\Gamma\curvearrowright (X,\mu)$ be a measure preserving action of a countable group $\Gamma$ on a standard probability space $(X,\mu)$. We prove that if the action $\Gamma\curvearrowright X$ is not profinite and satisfies a certain spectral gap condition, then there does not exist a countable-to-one Borel homomorphism from its orbit equivalence relation to the orbit equivalence relation of any modular action (i.e., an inverse limit of actions on countable sets). As a consequence, we show that if $\Gamma$ is a countable dense subgroup of a compact non-profinite group $G$ such that the left translation action $\Gamma\curvearrowright G$ has spectral gap, then $\Gamma\curvearrowright G$ is antimodular and not orbit equivalent to any, {\it not necessarily free}, profinite action. This provides the first such examples of compact actions, partially answering a question of Kechris and answering a question of Tsankov.