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

### Sandrine Caruso

Université Rennes 1, France### Bert Wiest

Université de Rennes I, France

## Abstract

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.

## Cite this article

Sandrine Caruso, Bert Wiest, On the genericity of pseudo-Anosov braids II: conjugations to rigid braids. Groups Geom. Dyn. 11 (2017), no. 2, pp. 549–565

DOI 10.4171/GGD/407