On universal continuous actions on the Cantor set

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.