Abelian subgroups of Garside groups

Abstract

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.

This article replaces our earlier preprint "Stable super summit sets in Garside groups", arXiv:math.GT/0602582

References in corpus (6)

• Non-positively curved aspects of Artin groups of finite type
• Garside groups are strongly translation discrete
• Translation numbers in a Garside group are rational with uniformly bounded denominators
• Some power of an element in a Garside group is conjugate to a periodically geodesic element
• Dimension and rank for mapping class groups
• The Novikov conjecture for mapping class groups as a corollary of Hamenstadt's theorem

Cited by in corpus (6)

• A Garside-theoretic approach to the reducibility problem in braid groups
• Translation numbers in a Garside group are rational with uniformly bounded denominators
• Some power of an element in a Garside group is conjugate to a periodically geodesic element
• Periodic elements in Garside groups
• Conjugacy classes of periodic braids
• Noncrossing partitions for periodic braids