Are large distance Heegaard splittings generic ?

TL;DR

This paper demonstrates that high-distance irreducible genus g Heegaard splittings are generically prevalent among all such splittings, using a new intrinsic notion of genericity based on measure theory.

Contribution

It introduces a novel measure-theoretic notion of genericity for curves in the curve complex and applies it to show high-distance Heegaard splittings are typical.

Findings

High-distance irreducible genus g Heegaard splittings are generic.

The new notion of genericity is based on Lebesgue measure on projective measured laminations.

The results differ from those obtained via random walk methods.

Abstract

In a previous paper we introduced a notion of "genericity" for countable sets of curves in the curve complex of a surface S, based on the Lebesgue measure on the space of projective measured laminations in S. With this definition we prove that for each fixed g > 1 the set of irreducible genus g Heegaard splittings of high distance is generic, in the set of all irreducible Heegaard splittings. Our definition of "genericity" is different and more intrinsic then the one given via random walks.