Geodesics in the braid group on three strands
Group Theory
2010-04-05 v1
Abstract
We study the geodesic growth series of the braid group on three strands, B_3 := <a,b|aba = bab>. We show that the set of geodesics of B_3 with respect to the generating set S := {a,b,a^-1,b^-1} is a regular language, and we provide an explicit computation of the geodesic growth series with respect to this set of generators. In the process, we give a necessary and sufficient condition for a freely reduced word w in S^* to be geodesic in B_3 with respect to S. Also, we show that the translation length with respect to S of any element in B_3 is an integer.
Cite
@article{arxiv.math/0311153,
title = {Geodesics in the braid group on three strands},
author = {Lucas Sabalka},
journal= {arXiv preprint arXiv:math/0311153},
year = {2010}
}
Comments
18 pages, 3 figures. To be published in Proceedings of the AMS