Periodic elements in Garside groups

Let G be a Garside group with Garside element Δ, and let Δm be the minimal positive central power of Δ. An element g∈G is said to be periodic if some power of it is a power of Δ. In this paper, we study periodic elements in Garside groups and their conjugacy classes.We show that the periodicity of an element does not depend on the choice of a particular Garside structure if and only if the center of G is cyclic; if gk=Δka for some nonzero integer k, then g is conjugate to Δa; every finite subgroup of the quotient group G/〈Δm〉 is cyclic.By a classical theorem of Brouwer, Kerékjártó and Eilenberg, an n-braid is periodic if and only if it is conjugate to a power of one of two specific roots of Δ2. We generalize this to Garside groups by showing that every periodic element is conjugate to a power of a root of Δm.We introduce the notions of slimness and precentrality for periodic elements, and show that the super summit set of a slim, precentral periodic element is closed under any partial cycling. For the conjugacy problem, we may assume the slimness without loss of generality. For the Artin groups of type , , , and the braid group of the complex reflection group of type (e,e,n), endowed with the dual Garside structure, we may further assume the precentrality.