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

Sandrine Caruso (1) , Bert Wiest (1)

1 IRMAR - Institut de Recherche Mathématique de Rennes

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 $\ge$ 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.

Mots clés: théorie des tresses; généricité; problème de conjugaison

Domaines: Topologie géométrique [math.GT]

Sandrine Caruso, Bert Wiest. On the genericity of pseudo-Anosov braids II: conjugations to rigid braids. Groups, Geometry, and Dynamics, 2017, 11 (2), pp.549-565. ⟨10.4171/GGD/407⟩. ⟨hal-00865312v2⟩