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

We show that for each element g of a Garside group, there existsa positive integer m such that g^m is conjugate to a periodicallygeodesic element h, an element with |hⁿ| = |n| · |h|for all integers n, where |g| denotes the shortest word lengthof g with respect to the set of simple elements. We also showthat there is a finite-time algorithm that computes, given anelement of a Garside group, its stable super summit set.