Bounded cohomology of subgroups of mapping class groups

Abstract

We show that every subgroup of the mapping class group $MCG\left(S\right)$ of a compact surface $S$ is either virtually abelian or it has infinite dimensional second bounded cohomology. As an application, we give another proof of the Farb–Kaimanovich–Masur rigidity theorem that states that $MCG\left(S\right)$ does not contain a higher rank lattice as a subgroup.