# On a family of unitary representations of mapping class groups

## TL;DR

This paper introduces a new family of unitary representations of mapping class groups based on measured foliations, demonstrating their lack of almost invariant vectors and deriving related inequalities and classifications.

## Contribution

It presents a novel family of unitary representations for Mod(S) and analyzes their properties, including invariance and classification up to weak equivalence.

## Key findings

- None of the representations has almost invariant vectors.
- An inequality for the action of Mod(S) on Teichmüller space is established.
- A classification of unitary quasi-representations up to weak equivalence is provided.

## 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.