The conjugacy problem and virtually cyclic subgroups in the Artin braid group quotient $B_n/[P_n,P_n]$

TL;DR

This paper addresses the conjugacy problem in a specific quotient of the Artin braid group by developing algebraic techniques and explicitly constructing infinite virtually cyclic subgroups.

Contribution

It introduces a novel approach using systems of equations over integers to solve the conjugacy problem in the quotient group and explicitly constructs virtually cyclic subgroups.

Findings

Conjugacy problem is solvable in the quotient group.

Explicit realization of infinite virtually cyclic subgroups.

New algebraic techniques for analyzing braid group quotients.

Abstract

Let $n\geq 3$. In this paper we deal with the conjugacy problem in the Artin braid group quotient $B_n/[P_n,P_n]$. To solve it we use systems of equations over the integers arising from the action of $B_n/[P_n,P_n]$ over the abelianization of the pure Artin braid group $P_n/[P_n,P_n]$. Using this technique we also realize explicitly infinite virtually cyclic subgroups in $B_n/[P_n,P_n]$.