The R$_\infty$ property for pure Artin braid groups
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2020
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| _version_ | 1866918187052826624 |
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| author | Dekimpe, Karel Gonçalves, Daciberg Lima Ocampo, Oscar |
| author_facet | Dekimpe, Karel Gonçalves, Daciberg Lima Ocampo, Oscar |
| contents | 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 $\infty$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2006_08286 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | The R$_\infty$ property for pure Artin braid groups Dekimpe, Karel Gonçalves, Daciberg Lima Ocampo, Oscar Group Theory Representation Theory Primary: 20E36, Secondary: 20F36, 20E45, 20C30 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 $\infty$. |
| title | The R$_\infty$ property for pure Artin braid groups |
| topic | Group Theory Representation Theory Primary: 20E36, Secondary: 20F36, 20E45, 20C30 |
| url | https://arxiv.org/abs/2006.08286 |