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arXiv · 1803.05461

On universal continuous actions on the Cantor set

Gábor Elek
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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.

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Gábor Elek. 2018-03-14. On universal continuous actions on the Cantor set. https://arxiv.org/abs/1803.05461

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