On a structure of random open books and closed braids

TL;DR

This paper explores how random walks in the mapping class group influence the properties of 3-manifolds and links, demonstrating that random closed braids and open books tend to be hyperbolic, with implications for their geometric structures.

Contribution

It establishes new connections between random walks in the mapping class group and the hyperbolicity of random 3-manifolds and links, extending previous results on fractional Dehn twist coefficients.

Findings

Random closed braids are hyperbolic.

Random open books are hyperbolic.

Large fractional Dehn twist coefficients influence manifold properties.

Abstract

A result of Malyutin shows that a random walk on the mapping class group gives rise to an element whose fractional Dehn twist coefficient is large or small enough. We show that this leads to several properties of random 3-manifolds and links. For example, random closed braids and open books are hyperbolic.