Linear representations of the mapping class group of dimension at most $3g-3$

TL;DR

This paper classifies low-dimensional linear representations of the mapping class group of a surface, showing they are essentially built from known geometric actions plus trivial parts, with implications for algebraic structures on moduli spaces.

Contribution

It provides a complete classification of representations up to dimension 3g-3, revealing their structure as sums of geometric and trivial components.

Findings

Representations are direct sums of geometric and trivial parts.

Any such representation is either in dimension 2g or 2g+1.

Linear systems on moduli spaces in this range are of algebro-geometric origin.

Abstract

We classify representations of the mapping class group of a surface of genus $g$ (with at most one puncture or boundary component) up to dimension $3g-3$. Any such representation is the direct sum of a representation in dimension $2g$ or $2g+1$ (given as the action on the (co)homology of the surface or its unit tangent bundle) with a trivial representation. As a corollary, any linear system on the moduli space of Riemann surfaces of genus $g$ in this range is of algebro-geometric origin.