Full groups of minimal homeomorphisms

TL;DR

This paper investigates the algebraic and topological properties of full groups arising from minimal homeomorphisms on Cantor spaces, revealing their non-Polish nature, complexity, and simplicity.

Contribution

It demonstrates that these full groups lack compatible Polish topologies, are coanalytic non-Borel, and are topologically simple, providing new insights into their structure.

Findings

Full groups do not admit compatible Polish group topologies.

Full groups of minimal homeomorphisms are coanalytic non-Borel.

These groups are topologically simple.

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 $\Z$-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 $\Homeo(X)$.