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arXiv · 2006.08286

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

Karel Dekimpe·Daciberg Lima Gonçalves·Oscar Ocampo
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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 $\operatorname{\text{Aut}}(P_n) \to \operatorname{\text{Aut}}(Γ_2 (P_n)/Γ_3(P_n))$ which turns out to factor through a representation $ρ\colon S_{n+1} \to \operatorname{\text{Aut}}(Γ_2 (P_n)/Γ_3(P_n))$. We can then use representation theory of the symmetric groups to show that any automorphism $α$ of $P_n$ acts on the free abelian group $Γ_2 (P_n)/Γ_3(P_n)$ via a matrix with an eigenvalue equal to 1. This allows us to conclude that the Reidemeister number $R(α)$ of $α$ is $\infty$.

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Karel Dekimpe, Daciberg Lima Gonçalves, Oscar Ocampo. 2020-06-15. The R$_\infty$ property for pure Artin braid groups. https://doi.org/10.1007/s00605-020-01484-7

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