Mapping class groups of medium distance Heegaard splittings

TL;DR

This paper demonstrates that for Heegaard splittings with Hempel distance greater than three, the associated mapping class group is a subgroup of the ambient manifold's mapping class group, leading to finiteness results for certain automorphism groups.

Contribution

It establishes a new relationship between the mapping class groups of high-distance Heegaard splittings and the ambient manifold, extending understanding of their algebraic structure.

Findings

Mapping class group is isomorphic to a subgroup of the ambient manifold's group for distance > 3.

Automorphism group of the curve complex preserving two handlebody sets is finite when they are distance ≥ 4.

Provides new insights into the structure of Heegaard splittings with large Hempel distance.

Abstract

We show that if the Hempel distance of a Heegaard splitting is larger than three then the mapping class group of the Heegaard splitting is isomorphic to a subgroup of the mapping class group of the ambient 3-manifold. This implies that given two handlebody sets in the curve complex for a surface that are distance at least four apart, the group of automorphisms of the curve complex that preserve both handlebody sets is finite.