Crossed homomorphisms and low dimensional representations of mapping class groups of surfaces

TL;DR

This paper classifies low-dimensional complex linear representations of pure mapping class groups of surfaces for genus g ≥ 7, showing no irreducible representations of dimension 2g+1 exist in this range.

Contribution

It provides a complete classification of (2g+1)-dimensional representations for g ≥ 7 using twisted 1-cohomology, extending previous studies.

Findings

No irreducible (2g+1)-dimensional representations for g ≥ 7

Classification based on twisted 1-cohomology groups

Contrast with known results for g=2

Abstract

We continue the study of low dimensional linear representations of mapping class groups of surfaces initiated by Franks--Handel and Korkmaz. We consider $(2g+1)$-dimensional complex linear representations of the pure mapping class groups of compact orientable surfaces of genus $g$. We give a complete classification of such representations for $g \geq 7$ up to conjugation, in terms of certain twisted $1$-cohomology groups of the mapping class groups. A new ingredient is to use the computation of a related twisted $1$-cohomology group by Morita. The classification result implies in particular that there are no irreducible linear representations of dimension $2g+1$ for $g \geq 7$, which marks a contrast with the case $g=2$.