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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.doi:10.2140/gt.2012.16.111 fatcat:gpwhiumbmndjjfggehcrbjbbeu