Horizontal Dehn Surgery and genericity in the curve complex

TL;DR

This paper introduces a new notion of genericity for subsets of the curve complex and demonstrates that certain curves leading to high-distance Heegaard splittings are generically prevalent in the set of all essential curves.

Contribution

It defines an intrinsic notion of genericity for the curve complex and applies it to show that curves producing high-distance Heegaard splittings are generic.

Findings

High-distance Heegaard splitting curves are generic in the set of all essential curves.

The new notion of genericity differs from existing probabilistic or limit-based definitions.

The approach applies to any 3-manifold M and its Heegaard surface.

Abstract

We introduce a general notion of "genericity" for countable subsets of a space with Borel measure, and apply it to the set of vertices in the curve complex of a surface S, interpreted as subset of the space of projective measured laminations in S, equipped with its natural Lebesgue measure. We prove that, for any 3-manifold M, the set of curves c on a Heegaard surface S in M, such that every non-trivial Dehn twist at c yields a Heegaard splitting of high distance, is generic in the set of all essential simple closed curves on S. Our definition of "genericity" is different and more intrinsic than alternative such existing notions, given e.g. via random walks or via limits of quotients of finite sets.