Notes on periodic elements of Garside groups

Abstract

Let $G$ be a Garside group with Garside element $\Delta$. An element $g$ in $G$ is said to be \emph{periodic} if some power of $g$ lies in the cyclic group generated by $\Delta$. This paper shows the following. (i) 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. (ii) If $g^k=\Delta^{ka}$ for some nonzero integer $k$, then $g$ is conjugate to $\Delta^a$. (iii) Every finite subgroup of the quotient group $G/<\Delta^m>$ is cyclic, where $\Delta^m$ is the minimal positive central power of $\Delta$.