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

TL;DR

This paper demonstrates that within the braid group, a positive proportion of braids are rigid pseudo-Anosov, regardless of the size of the considered subset, highlighting their genericity.

Contribution

It establishes a uniform lower bound on the proportion of rigid pseudo-Anosov braids in the braid group Cayley graph.

Findings

Positive proportion of rigid pseudo-Anosov braids in the braid group

Lower bound independent of the size of the subset

Highlights the genericity of pseudo-Anosov braids

Abstract

We prove that, in the $l$-ball of the Cayley graph of the braid group with $n \geqslant 3$ strands, the proportion of rigid pseudo-Anosov braids is bounded below independently of $l$ by a positive value.