On a family of unitary representations of mapping class groups

Abstract

For a compact surface $S = S_{g,n}$ with $3g + n \geq 4$, we introduce a family of unitary representations of the mapping class group Mod($S$) based on the space of measured foliations. For this family of representations, we show that none of them has almost invariant vectors. As one application, we obtain an inequality concerning the action of Mod($S$) on the Teichm\"{u}ller space of $S$. Moreover, using the same method plus recent results about weak equivalence, we also give a classification, up to weak equivalence, for the unitary quasi-representations with respect to geometrical subgroups.