Smallest nonabelian quotients of surface braid groups

TL;DR

This paper establishes the minimal possible size of nonabelian quotients of surface braid groups and classifies all such quotients that achieve this bound, revealing they are either symmetric groups or specific nilpotent p-groups.

Contribution

It provides a precise lower bound on nonabelian quotient sizes and classifies all minimal quotients for surface braid groups based on genus and strand number.

Findings

Minimal nonabelian quotients are either symmetric groups or 2-step nilpotent p-groups.

Classified all quotients attaining the minimal size depending on parameters.

Established sharp bounds for nonabelian quotients of surface braid groups.

Abstract

We give a sharp lower bound on the size of nonabelian quotients of the surface braid group $B_n(\Sigma_g)$ and classify all quotients that attain the lower bound: Depending on $n$ and $g$, a quotient of minimum order is either a symmetric group or a 2-step nilpotent $p$-group.