The $R_\infty$ property for pure Artin braid groups

Abstract

In this paper we prove that all pure Artin braid groups $P_n$ ($n\ge 3$) have the $R_\infty$ property. In order to obtain this result, we analyse the naturally induced morphism ${\text {Aut}}\left( {P_n}\right) \longrightarrow {\text {Aut}}\left( {\Gamma _2 (P_n)/\Gamma _3(P_n)}\right)$ which turns out to factor through a representation $\rho :S_{{n+1}} \longrightarrow {\text {Aut}}\left( {\Gamma _2 (P_n)/\Gamma _3(P_n)}\right)$. 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 $\infty$.