On finite quotients of surface braid groups having order at most $127$

TL;DR

This paper classifies all finite quotients of the pure braid group on two strands over a genus b surface with order at most 127 that are 'admissible', meaning they do not factor through the fundamental group of the surface product.

Contribution

It provides a complete classification of admissible finite quotients of the surface braid group with order up to 127, a novel result in the study of surface braid group quotients.

Findings

Identified all admissible quotients of order ≤ 127

Established criteria for admissibility in surface braid groups

Extended understanding of finite quotients in geometric group theory

Abstract

Let $\Sigma_b$ be a compact Riemann surface of genus $b \geq 2$ and let $\mathsf{P}_2(\Sigma_b)=\pi_1(\Sigma_b \times \Sigma_b - \Delta)$ be the corresponding pure braid group on two strands. A finite quotient $\varphi \colon \mathsf{P}_2(\Sigma_b) \to G$ is called "admissible" if $\varphi$ does not factor through $\pi_1(\Sigma_b \times \Sigma_b)$. In this work we classify all admissible quotients of $\mathsf{P}_2(\Sigma_b)$ such that $|G| \leq 127$.