Coxeter-type quotients of surface braid groups

TL;DR

This paper investigates Coxeter-type quotients of surface braid groups, focusing on their algebraic structure when modded out by specific relations involving the standard Artin generators.

Contribution

It introduces new quotient groups of surface braid groups by analyzing relations involving the Artin generator and the pure braid subgroup, extending understanding of their algebraic properties.

Findings

Characterization of Coxeter-type quotients for closed surfaces and disks.

Analysis of the impact of the relation ^q=1 on group structure.

Insights into the algebraic structure of these quotient groups.

Abstract

Let $M$ be a closed surface, $q\geq 2$ and $n\geq 2$. In this paper, we analyze the Coxeter-type quotient group $B_n(M)(q)$ of the surface braid group $B_{n}(M)$ by the normal closure of the element $\sigma_1^q$, where $\sigma_1$ is the classic Artin generator of the Artin braid group $B_n$. Also, we study the Coxeter-type quotient groups obtained by taking the quotient of $B_n(M)$ by the commutator subgroup of the respective pure braid group $[P_n(M),P_n(M)]$ and adding the relation $\sigma_1^q=1$, when $M$ is a closed orientable surface or the disk.