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 \(\infty \).