Finite abelian subgroups of the mapping class group

Abstract

The problem of enumeration of conjugacy classes of finite abelian subgroups of the mapping class group ${\mathcal{ℳ}}_{\sigma }$ of a closed, smooth, orientable surface $S$ of genus $\sigma \ge 2$ is considered. A complete method of enumeration is achieved for finite elementary abelian subgroups and steps are taken toward enumeration of finite abelian subgroups.