Twisted conjugacy in braid groups

TL;DR

This paper presents an algorithm to solve the twisted conjugacy problem in braid groups, enabling the decision of conjugacy relations under automorphisms, and extends results to certain semidirect product groups.

Contribution

It introduces a novel algorithm for the twisted conjugacy problem in braid groups and applies it to show solvability of the conjugacy problem in specific semidirect product groups.

Findings

Algorithm solves the twisted conjugacy problem in braid groups.

Decides conjugacy under automorphisms for braids.

Shows solvability of conjugacy problem in certain semidirect products.

Abstract

In this note we solve the twisted conjugacy problem for braid groups, i.e. we propose an algorithm which, given two braids $u,v\in B_n$ and an automorphism $\phi \in Aut (B_n)$, decides whether $v=(\phi (x))^{-1}ux$ for some $x\in B_n$. As a corollary, we deduce that each group of the form $B_n \rtimes H$, a semidirect product of the braid group $B_n$ by a torsion-free hyperbolic group $H$, has solvable conjugacy problem.