Random Heegaard splittings

TL;DR

This paper studies the properties of 3-manifolds obtained from random Heegaard splittings, showing they are hyperbolic with high probability and analyzing the growth of certain geometric measures.

Contribution

It demonstrates the asymptotic independence of random walk distributions and proves linear growth of translation length and curve complex distance in random Heegaard splittings.

Findings

Joint distribution of random walks and their inverses becomes independent

Translation length on the curve complex grows linearly with walk length

Distance between disc sets in the curve complex grows linearly with n

Abstract

A random Heegaard splitting is a 3-manifold obtained by using a random walk of length n on the mapping class group as the gluing map between two handlebodies. We show that the joint distribution of random walks of length n and their inverses is asymptotically independent, and converges to the product of the harmonic and reflected harmonic measures defined by the random walk. We use this to show that the translation length on the curve complex of a random walk grows linearly in the length of the walk, and similarly, that distance in the curve complex between the disc sets of a random Heegaard splitting grows linearly in n. In particular, this implies that a random Heegaard splitting is hyperbolic with asymptotic probability one.