Small generating sets for the Torelli group

TL;DR

This paper proves that the Torelli subgroup of the mapping class group has a finite generating set whose size grows cubically with the genus, using a new handle graph space for the group action.

Contribution

It establishes the first explicit cubic bound on the size of generating sets for the Torelli group, confirming a conjecture of Dennis Johnson.

Findings

Finite generating set size grows cubically with genus

Introduction of the handle graph for group action analysis

Proof of Johnson's conjecture on Torelli group generators

Abstract

Proving a conjecture of Dennis Johnson, we show that the Torelli subgroup of the mapping class group has a finite generating set whose size grows cubically with respect to the genus of the surface. Our main tool is a new space called the handle graph on which the Torelli group acts cocompactly.