Notes on periodic elements of Garside groups

Details Notes on periodic elements of Garside groups Notes on periodic elements of Garside groups

2008-08-03

arxiv.org/abs/0808.0308v3

Mathematics

Geometric Topology

The contents of this 8-page paper have been subsumed into the 27-page paper, "Periodic elements in Garside groups" (arXiv:1004

Scientific paper

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 $\Delta$.

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