Full groups, flip conjugacy, and orbit equivalence of Cantor minimal systems

Abstract

In the paper, we consider the full group $[\varphi ]$ and topological full group $[[\varphi ]]$ of a Cantor minimal system $(X,\f)$. We prove that the commutator subgroups $D([\f])$ and $D([[\f]])$ are simple and show that the groups $D([\f])$ and $D([[\f]])$ completely determine the class of orbit equivalence and flip conjugacy of $\f$, respectively. These results improve the classification found in \cite{gps:1999}. As a corollary of the technique used, we establish the fact that $\f$ can be written as a product of three involutions from $[\f]$.

17 pages, references added, some typos fixed