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Geodesics in the braid group on three strands

Lucas Sabalka
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Abstract

We study the geodesic growth series of the braid group on three strands, B_3 := . 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.

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Lucas Sabalka. 2003-11-10. Geodesics in the braid group on three strands. https://arxiv.org/abs/math/0311153

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