Heegaard Floer homology and the word metric on the Torelli group

TL;DR

This paper explores the connection between Heegaard Floer homology correction terms and the word metric on the Torelli group, revealing properties of the group's geometry and its subsets.

Contribution

It provides an elementary proof of the infinite diameter of the Torelli group's Cayley graph and analyzes bounded subsets within the group.

Findings

Cayley graph of Torelli group has infinite diameter

Many subsets of Torelli group are bounded

Ruling out Morita-type formulas for certain subgroups

Abstract

We study a relationship between the Heegaard Floer homology correction terms of integral homology spheres and the word metric on the Torelli group. For example, we give an elementary proof that the Cayley graph of the Torelli group has infinite diameter in the word metric induced by the generating set of all separating twists and bounding pair maps. On the other hand, we show that many subsets of the Torelli group are bounded with respect to this metric. Finally, we address the case of rational homology spheres by ruling out a certain Morita-type formula for congruence subgroups of mapping class groups.