Full groups of minimal homeomorphisms

Abstract

We study full groups of minimal actions of countable groups by homeomorphisms on a Cantor space $X$, showing that these groups do not admit a compatible Polish group topology and, in the case of $\mathbb{Z}$-actions, are coanalytic non-Borel inside $\Homeo(X)$. We point out that the full group of a minimal homeomorphism is topologically simple. We also study some properties of the closure of the full group of a minimal homeomorphism inside $\Homeo(X)$.

References in corpus (3)

• Full Groups and Orbit Equivalence in Cantor Dynamics
• Random Orderings and Unique Ergodicity of Automorphism Groups
• Cantor systems, piecewise translations and simple amenable groups