Some power of an element in a Garside group is conjugate to a periodically geodesic element

Abstract

We show that for each element $g$ of a Garside group, there exists a positive integer $m$ such that $g^m$ is conjugate to a periodically geodesic element $h$, an element with $|h^n|_\D=|n|\cdot|h|_\D$ for all integers $n$, where $|g|_\D$ denotes the shortest word length of $g$ with respect to the set $\D$ of simple elements. We also show that there is a finite-time algorithm that computes, given an element of a Garside group, its stable super summit set.

arXiv commentSubj-class of this paper should be Geometric Topology; Version published by BLMS

References in corpus (6)

• Non-positively curved aspects of Artin groups of finite type
• Garside groups are strongly translation discrete
• Translation numbers in a Garside group are rational with uniformly bounded denominators
• Abelian subgroups of Garside groups
• Stable super summit sets in Garside groups
• Small cancellation groups and translation numbers

Cited by in corpus (10)

• A Garside-theoretic approach to the reducibility problem in braid groups
• Reducible braids and Garside theory
• Braid groups and Artin groups
• Translation numbers in a Garside group are rational with uniformly bounded denominators
• Abelian subgroups of Garside groups
• Periodic elements in Garside groups
• General cycling operations in Garside groups
• A polynomial-time solution to the reducibility problem
• Conjugacy classes of periodic braids
• Notes on periodic elements of Garside groups