Finite orbits of Hurwitz actions on braid systems

Abstract

There are natural actions of the braid group $B_{n}$ on $B_{m}^{n}$, the $n$-fold product of the braid group $B_{m}$, 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 to be finite and give an upper bound of the size for a finite orbit in $n=2$ or $m=3$ case.