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 gm is conjugate to a periodically geodesic element h, an element with |hn|𝒟 = |n| · |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.