Uniqueness of roots up to conjugacy for some affine and finite type Artin groups

TL;DR

This paper proves that in certain Artin groups of finite and affine types, elements with equal powers are conjugate, extending known results for type A and using braid group classifications to establish root uniqueness and conjugacy properties.

Contribution

The paper generalizes the uniqueness of roots up to conjugacy to specific affine and finite type Artin groups, introducing a stronger theorem via braid group analysis.

Findings

Elements with equal nonzero powers are conjugate in specified Artin groups.

In pure braid groups, roots are unique up to conjugacy.

The conjugating element can be chosen with specific linking properties.

Abstract

Let $G$ be one of the Artin groups of finite type ${\mathbf B}_n={\mathbf C}_n$, and affine type $\tilde{\mathbf A}_{n-1}$ and $\tilde{\mathbf C}_{n-1}$. In this paper, we show that if $\alpha$ and $\beta$ are elements of $G$ such that $\alpha^k=\beta^k$ for some nonzero integer $k$, then $\alpha$ and $\beta$ are conjugate in $G$. For the Artin group of type $\mathbf A_n$, this was recently proved by J. Gonz\'alez-Meneses. In fact, we prove a stronger theorem, from which the above result follows easily by using descriptions of those Artin groups as subgroups of the braid group on $n+1$ strands. Let $P$ be a subset of $\{1,...,n\}$. An $n$-braid is said to be \emph{$P$-pure} if its induced permutation fixes each $i\in P$, and \emph{$P$-straight} if it is $P$-pure and it becomes trivial when we delete all the $i$-th strands for $i\not\in P$. Exploiting the Nielsen-Thurston classification of braids, we show that if $\alpha$ and $\beta$ are $P$-pure $n$-braids such that $\alpha^k=\beta^k$ for some nonzero integer $k$, then there exists a $P$-straight $n$-braid $\gamma$ with $\beta=\gamma\alpha\gamma^{-1}$. Moreover, if $1\in P$, the conjugating element $\gamma$ can be chosen to have the first strand algebraically unlinked with the other strands. Especially in case of $P=\{1,...,n\}$, our result implies the uniqueness of root of pure braids, which was known by V. G. Bardakov and by D. Kim and D. Rolfsen.