Small generating sets for the Torelli group

Abstract

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