Full groups of minimal homeomorphisms

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 $\text{Homeo}(X)$. 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 $\text{Homeo}(X)$.