# Linearity of some low-complexity mapping class groups

## TL;DR

This paper demonstrates that certain low-complexity pure mapping class groups of surfaces are linear by analyzing their presentations and relating them to braid and Artin groups.

## Contribution

It shows that pure mapping class groups of genus 0 and 1 surfaces with limited boundary and punctures are linear, connecting them to well-understood algebraic groups.

## Key findings

- Pure mapping class groups are isomorphic to braid and Artin groups in specific cases.
- These groups are proven to be linear in the analyzed cases.
- The results extend understanding of the algebraic structure of low-complexity surface groups.

## Abstract

By analyzing known presentations of the pure mapping groups of orientable surfaces of genus $g$ with $b$ boundary components and $n$ punctures, we show that these groups are isomorphic to some groups related to the braid groups and the Artin group of type $D_4$ in the cases when $g=0$ with $b$ and $n$ arbitrary, and when $g=1$ and $b+n$ is at most $3$. As a corollary, we conclude that the pure mapping class groups are linear in these cases.