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

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. González-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 $i\notin P$. 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.

15 pages, 8 figures; version published by Math. Z

References in corpus (3)

• The n-th root of a braid is unique up to conjugacy
• A Garside-theoretic approach to the reducibility problem in braid groups
• Conjugacy classes of periodic braids

Cited by in corpus (6)

• Braid groups of imprimitive complex reflection groups
• Periodic elements in Garside groups
• Endomorphisms of Artin groups of type $B_n$
• Uniqueness of roots up to conjugacy in circular and hosohedral-type Garside groups
• Notes on periodic elements of Garside groups
• Injectivity on the set of conjugacy classes of some monomorphisms between Artin groups