Periodic elements in Garside groups

Abstract

Let $G$ be a Garside group with Garside element $\Delta$, and let $\Delta^m$ be the minimal positive central power of $\Delta$. An element $g\in G$ is said to be 'periodic' if some power of it is a power of $\Delta$. 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 $g^k=\Delta^{ka}$ for some nonzero integer $k$, then $g$ is conjugate to $\Delta^a$; every finite subgroup of the quotient group $G/<\Delta^m>$ is cyclic. By a classical theorem of Brouwer, Ker\'ekj\'art\'o and Eilenberg, an $n$-braid is periodic if and only if it is conjugate to a power of one of two specific roots of $\Delta^2$. We generalize this to Garside groups by showing that every periodic element is conjugate to a power of a root of $\Delta^m$. 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 $A_n$, $B_n$, $D_n$, $I_2(e)$ 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.