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