Mapping class groups are not linear in positive characteristic

TL;DR

This paper proves that the mapping class group of certain surfaces cannot be faithfully represented as matrices over any field of positive characteristic, highlighting a fundamental limitation in their linearity.

Contribution

It establishes the non-linearity of mapping class groups in positive characteristic fields for surfaces of genus at least 3, extending understanding of their algebraic properties.

Findings

No faithful linear representations over positive characteristic fields

Applicable to surfaces with genus ≥ 3, including closed and punctured surfaces

Advances the understanding of algebraic constraints of mapping class groups

Abstract

For $\Sigma$ an orientable surface of finite topological type having genus at least 3 (possibly closed or possibly with any number of punctures or boundary components), we show that the mapping class group $Mod(\Sigma)$ has no faithful linear representation in any dimension over any field of positive characteristic.