论文标题
R $ _ \ infty $属性纯Artin辫子组
The R$_\infty$ property for pure Artin braid groups
论文作者
论文摘要
在本文中，我们证明所有纯Artin辫子组$ p_n $（$ n \ geq 3 $）均具有$ r_ \ infty $属性。为了获得此结果，我们分析了自然感应的形态$ \ operatorAtorname {\ text {aut}}}（p_n）\ to \ propatatorName {\ text {aut}}（γ_2（γ_2（p_n）/γ_3（p_n））$ colon $ s_ \ operatatorName {\ text {aut}}（γ_2（p_n）/γ_3（p_n））$。然后，我们可以使用对称组的表示理论表明，$ p_n $的任何自动形态$α$均在免费的Abelian group $γ_2（p_n）/γ_3（p_n）/γ_3（p_n）$上作用，其eigenvalue等于1。
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 $\infty$.
