Big Handlebody Distance Implies Finite Mapping Class Group

TL;DR

This paper proves that a large handlebody distance in a closed 3-manifold's Heegaard splitting implies the finiteness of its mapping class group, revealing a deep connection between geometric properties and algebraic structure.

Contribution

It establishes a new criterion linking handlebody distance to the finiteness of the mapping class group of a 3-manifold, advancing understanding of 3-manifold symmetries.

Findings

Large handlebody distance implies finite subgroup of the mapping class group

Finiteness of the mapping class group under certain geometric conditions

Extension of the result to the entire mapping class group of the manifold

Abstract

We show that if $M$ is a closed three manifold with a Heegaard splitting with sufficiently big "handlebody distance" then the subgroup of the mapping class group of the Heegaard surface, which extend to both handlebodies is finite. As a corollary, this implies that under the same hypothesis, the mapping class group of $M$ is finite.