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

Abstract

In the paper, we consider the full group \([\phi]\) and topological full group \([[\phi]]\) 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 \([\f]\).