Dynamics of subgroups of mapping class groups

Let $\rm{Mod(S)}$ be the mapping class group of a closed orientable surface $S$ of genus $g \geq 2$. Let $G$ be a non-elementary subgroup of $\rm{Mod(S)}$ so that the associated Bowen-Margulis measure is finite. In this paper, we give an asymptotic growth formula for $G$ with respect to the Teichm\"{u}ller metric.