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

TL;DR

The paper proves that the pure Artin braid group $P_n$ has the $R_\infty$ property for all $n\ge 3$ by passing to the quotient $\Gamma_2(\overline{P}_n)/\Gamma_3(\overline{P}_n)$ and exploiting a finite representation $\rho: S_{n+1}\to \mathrm{Aut}(\Gamma_2(\overline{P}_n)/\Gamma_3(\overline{P}_n))$. Representation theory of the symmetric group is used to show every automorphism acts with an eigenvalue $1$ on this free-abelian quotient, forcing the Reidemeister number to be infinite. The authors treat $n=3,4$ separately, with $P_3\cong F_2\times \mathbb{Z}$ and a central-series argument for $P_4$, and then handle $n\ge 5$ by identifying $\rho$ with the irreducible $S_{n+1}$-representation of shape $(n-3,1,1,1)$ and applying Stembridge’s theory. This establishes the $R_\infty$ property for all pure braid groups, extending known results from the full braid groups.

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 $\infty$.

Theorems & Definitions (20)

• Remark 1
• Lemma 2
• proof
• Proposition 3
• proof
• Lemma 4
• proof
• Proposition 5
• proof
• Theorem 6