The R$_\infty$ property for pure Artin braid groups

TL;DR

This paper proves that all pure Artin braid groups with at least three strands possess the $R_ty$ property, meaning every automorphism has an infinite Reidemeister number, using representation theory of symmetric groups.

Contribution

It establishes the $R_ty$ property for pure Artin braid groups by analyzing automorphisms through symmetric group representations and eigenvalue arguments.

Findings

All pure Artin braid groups $P_n$ ($n extgreater=3$) have the $R_ty$ property.

Automorphisms induce matrices with eigenvalue 1 on a specific abelian quotient.

Reidemeister number of any automorphism in these groups is infinite.

Abstract

In this paper we prove that all pure Artin braid groups $P_n$ ($n\geq 3$) have the $R_\infty$ property. In order to obtain this result, we analyse the naturally induced morphism $\operatorname{\text{Aut}}(P_n) \to \operatorname{\text{Aut}}(\Gamma_2 (P_n)/\Gamma_3(P_n))$ which turns out to factor through a representation $\rho\colon S_{n+1} \to \operatorname{\text{Aut}}(\Gamma_2 (P_n)/\Gamma_3(P_n))$. We can then use representation theory of the symmetric groups to show that any automorphism $\alpha$ of $P_n$ acts on the free abelian group $\Gamma_2 (P_n)/\Gamma_3(P_n)$ via a matrix with an eigenvalue equal to 1. This allows us to conclude that the Reidemeister number $R(\alpha)$ of $\alpha$ is $\infty$.