Mathematics – Geometric Topology
Scientific paper
2010-04-29
Journal of Pure and Applied Algebra, vol. 215, no. 10, pp. 2295-2314, 2011
Mathematics
Geometric Topology
The contents of the 8-page paper "Notes on periodic elements of Garside groups" (arXiv:0808.0308) have been subsumed into this
Scientific paper
10.1016/j.jpaa.2010.12.011
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.
Lee Eon-Kyung
Lee Sang-Jin
No associations
LandOfFree
Periodic elements in Garside groups does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with Periodic elements in Garside groups, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Periodic elements in Garside groups will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-285270