Notes on periodic elements of Garside groups

TL;DR

This paper investigates the properties of periodic elements in Garside groups, revealing conditions for their conjugacy, independence of Garside structure, and the cyclic nature of finite subgroups in certain quotients.

Contribution

It establishes when periodicity is structure-independent, characterizes conjugacy of elements related to the Garside element, and describes the structure of finite subgroups in specific quotients.

Findings

Periodicity independence occurs iff the center is cyclic.

Elements with powers equal to a central power are conjugate to that power.

Finite subgroups of certain quotients are cyclic.

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 $\Delta$.