Finite image homomorphisms of the braid group and its generalizations

TL;DR

This paper develops new methods using symmetric sets to analyze finite quotients of braid groups, improving bounds and extending results to virtual and welded braid groups.

Contribution

It introduces multiple symmetric set techniques to better estimate finite quotients and applies these to virtual and welded braid groups for the first time.

Findings

Improved superexponential lower bounds for braid group quotients

Extension of techniques to virtual and welded braid groups

Identification of symmetric sets in generalized braid groups

Abstract

Using totally symmetric sets, Chudnovsky, Kordek, Li, and Partin gave a superexponential lower bound on the cardinality of non-abelian finite quotients of the braid group. In this paper, we develop new techniques using multiple totally symmetric sets to count elements in non-abelian finite quotients of the braid group. Using these techniques, we improve the lower bound found by Chudnovsky et al. We exhibit totally symmetric sets in the virtual and welded braid groups, and use our new techniques to find superexponential bounds for the finite quotients of the virtual and welded braid groups.