On universal continuous actions on the Cantor set

TL;DR

The paper constructs a universal free continuous action of any countably infinite group on the Cantor set, capable of embedding all free Borel actions of the group on standard Borel spaces.

Contribution

It proves the existence of a universal free continuous action on the Cantor set for any countably infinite group, extending to nonfree actions with uniformly recurrent subgroups.

Findings

Existence of a universal free continuous action on the Cantor set.

Extension to nonfree Borel actions with uniformly recurrent subgroups.

Construction method using proper Cantor colorings.

Abstract

Using the notion of proper Cantor colorings we prove the following theorem. For any countably infinite group $\Gamma$, there exists a free continuous action $\zeta: \Gamma \curvearrowright C$ on the Cantor set, which is universal in the following sense: for any free Borel action $\alpha: \Gamma \curvearrowright X$ on the standard Borel space, there exists an injective Borel map $\Theta_\alpha: X\to C$ such that $\Theta_\alpha\circ \alpha=\zeta \circ \Theta_\alpha$. We extend our theorem for (nonfree) Borel $(\Gamma,Z)$-actions, where $Z$ is a uniformly recurrent subgroup.