A VECTOR FIELD APPROACH TO MAPPING CLASS ACTIONS

Abstract

We present a vector field method for showing that certain subgroups of the mapping class group $\Gamma$ of a Riemann surface of infinite topological type act properly discontinuously. We apply the method to the group of homotopy classes of quasiconformal self-maps of the complement $\Omega$ of a Cantor set in $\mathbb{C}$. When the Cantor set has bounded geometric type, we show that $\Gamma(\Omega)$ acts on the Teichmüller space $T(\Omega)$ properly discontinuously. Also, we apply the same method to show that the pure mapping class group $\Gamma_0(\Omega \cup \{\infty\})$ acts properly discontinuously on $T(\Omega \cup \{\infty\})$.