Dynamics of subgroups of mapping class groups

Let $\mathrm{M}\mathrm{o}\mathrm{d}(\mathrm{S})$ be the mapping class group of a closed orientable surface $S$ of genus $g\ge 2$. Let $G$ be a non-elementary subgroup of $\mathrm{M}\mathrm{o}\mathrm{d}(\mathrm{S})$ so that the associated Bowen-Margulis measure is finite. In this paper, we give an asymptotic growth formula for $G$ with respect to the Teichmüller metric.