On the linearity of certain mapping class groups

TL;DR

This paper demonstrates that certain mapping class groups, including those of punctured spheres and genus 2 surfaces, are linear by leveraging the known linearity of braid groups and hyperelliptic groups.

Contribution

It extends the linearity property from braid groups to broader classes of mapping class groups, including hyperelliptic and genus 2 surface groups.

Findings

Mapping class groups of punctured spheres are linear.

Hyperelliptic mapping class groups are linear.

Genus 2 surface mapping class group is linear.

Abstract

S. Bigelow proved that the braid groups are linear. That is, there is a faithful representation of the braid group into the general linear group of some field. Using this, we deduce from previously known results that the mapping class group of a sphere with punctures and hyperelliptic mapping class groups are linear. In particular, the mapping class group of a closed orientable surface of genus 2 is linear.