On the genericity of pseudo-Anosov braids I: rigid braids

Sandrine Caruso
2017 Groups, Geometry, and Dynamics  
We prove that, in the l-ball of the Cayley graph of the braid group with n 3 strands, the proportion of rigid pseudo-Anosov braids is bounded below independently of l by a positive value. Proposition 2.1. The set of simple braids is in bijection with the set S n of permutations of n elements, via the canonical projection from B n to S n . Definition 2.2 (left-weighting). Let s 1 , s 2 be two simple braids in B n . We say that s 1 and s 2 are left-weighted, or that the pair (s 1 , s 2 ) is
more » ... 1 , s 2 ) is left-weighted, if there does not exist any generator σ i such that s 1 σ i and σ −1 i s 2 are both still simple. Definition 2.3 (starting set, finishing set). Let s ∈ B n be a simple braid. We call starting set of s the set S(s) = {i, σ i s} and finishing set of s the set F (s) = {i, s σ i }. Remark 2.4. Two simple braids s 1 and s 2 are left-weighted if and only if S(s 2 ) ⊂ F (s 1 ).
doi:10.4171/ggd/406 fatcat:qnu5m6kc5vaxlflwiscktgrtaa