Finite orbits of Hurwitz actions on braid systems

TL;DR

This paper investigates the conditions under which Hurwitz actions on braid systems have finite orbits, providing criteria and bounds based on the structure of roots in braid groups.

Contribution

It introduces a criterion for finiteness of Hurwitz orbits and establishes upper bounds for specific cases, advancing understanding of braid group actions.

Findings

Criteria for finite Hurwitz orbits

Upper bounds for orbits of length 2 and degree 3

Structural analysis of roots in braid groups

Abstract

There are natural actions of the braid groups on the products of the braid groups, called the Hurwitz action. We first study the roots of centralizers in the braid groups. By using the structure of the roots, we provide a criterion for the Hurwitz orbit become finite and give an upper bound of the size of a finite orbit in length 2 or degree 3 case.