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 $\mathrm{Aut}(P_n)\to \mathrm{Aut}({\Gamma }_2(P_n)/{\Gamma }_3(P_n))$ which turns out to factor through a representation $\rho :S_{n+1}\to \mathrm{Aut}({\Gamma }_2(P_n)/{\Gamma }_3(P_n))$. 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 $\infty$.

18 pages