Abelian subgroups of Garside groups

In this paper, we show that for every abelian subgroup $H$ of a Garside group, some conjugate $g^{-1}Hg$ consists of ultra summit elements and the centralizer of $H$ is a finite index subgroup of the normalizer of $H$. Combining with the results on translation numbers in Garside groups, we obtain an easy proof of the algebraic flat torus theorem for Garside groups and solve several algorithmic problems concerning abelian subgroups of Garside groups.Comment: This article replaces our earlier preprint "Stable super summit sets in Garside groups", arXiv:math.GT/060258

Double coset problem for parabolic subgroups of braid groups

We provide the first solution to the double coset problem (DCP) for a large class of natural subgroups of braid groups, namely for all parabolic subgroups which have a connected associated Coxeter graph. Update: We succeeded to solve the DCP for all parabolic subgroups of braid groups.Comment: 8 pages. Update remark adde

Periodic elements in Garside groups

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 $G/$ 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.Comment: The contents of the 8-page paper "Notes on periodic elements of Garside groups" (arXiv:0808.0308) have been subsumed into this version. 27 page

Translation numbers in a Garside group are rational with uniformly bounded denominators

It is known that Garside groups are strongly translation discrete. In this paper, we show that the translation numbers in a Garside group are rational with uniformly bounded denominators and can be computed in finite time. As an application, we give solutions to some group-theoretic problems.Comment: 12 pages, to appear in J. Pure Appl. Algebr

From braid groups to mapping class groups

This paper is a survey of some properties of the braid groups and related groups that lead to questions on mapping class groups