Dynamics of subgroups of mapping class groups

Abstract Let Mod ( S ) $\operatorname {\mathrm {Mod}}(S)$ upper M o d left parenthesis upper S right parenthesis be the mapping class group of a closed orientable surface S of genus g ≥ 2 $g \geq 2$ g greater than or equals 2 . Let G be a non-elementary subgroup of Mod ( S ) $\operatorname {\mathrm {Mod}}(S)$ upper M o d left parenthesis upper S right parenthesis such 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.