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(2009 / vol.11 / no.1)
Discreteness properties of translation numbers in Garside groups: Extended Abstract
Eon-Kyung Lee, Sang-Jin Lee
Pages. 53-61   



The translation number of an element in a combinatorial group
is defined as the asymptotic word length of the element.
The discreteness properties of translation numbers
have been studied for geometric groups such as biautomatic groups
and hyperbolic groups.

The Garside group is a lattice-theoretic generalization
of braid groups and Artin groups of finite type.
In this extended abstract, we show that the discreteness properties
of translation numbers in Garside groups are as good as
in hyperbolic groups:
(i) translation numbers of elements in a Garside group
are rational with uniformly bounded denominators;
(ii) for every element $g$ of a Garside group,
some power $g^m$ is conjugate to a periodically geodesic element.



1. Introduction
2. Ideas of our approach



Garside group, braid group, translation number, periodically geodesic



20F36, 20F10