Groups, Geometry, and Dynamics


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Volume 11, Issue 2, 2017, pp. 549–565
DOI: 10.4171/GGD/407

Published online: 2017-06-26

On the genericity of pseudo-Anosov braids II: conjugations to rigid braids

Sandrine Caruso[1] and Bert Wiest[2]

(1) Université Rennes 1, France
(2) Université de Rennes I, France

We prove that generic elements of braid groups are pseudo-Anosov, in the following sense: in the Cayley graph of the braid group with $n \geqslant 3$ strands, with respect to Garside's generating set, we prove that the proportion of pseudo-Anosov braids in the ball of radius $l$ tends to $1$ exponentially quickly as $l$ tends to infinity. Moreover, with a similar notion of genericity, we prove that for generic pairs of elements of the braid group, the conjugacy search problem can be solved in quadratic time. The idea behind both results is that generic braids can be conjugated „easily" into a rigid braid.

Keywords: Braid group, mapping class group, Garside group, pseudo-Anosov, rigid braid

Caruso Sandrine, Wiest Bert: On the genericity of pseudo-Anosov braids II: conjugations to rigid braids. Groups Geom. Dyn. 11 (2017), 549-565. doi: 10.4171/GGD/407