An infinite presentation of the Torelli group - [abstract] [pdf] [ps]
Geom. Funct. Anal. 19 (2009), no. 2, 591-643.

Abstract :
In this paper, we construct a presentation of the Torelli subgroup of the mapping class group of a surface whose generators consist of the set of all ``separating twists'', all ``bounding pair maps'', and all ``commutators of simply intersecting pairs'' and whose relations all come from a short list of topological configurations of these generators on the surface. Aside from a few obvious ones, all of these relations come from a set of embeddings of groups derived from surface groups into the Torelli group. In the process of analyzing these embeddings, we derive a novel presentation for the fundamental group of a closed surface whose generating set is the set of {\em all} simple closed curves. Our main tool for analyzing the Torelli group is a new theorem which allows us to obtain presentations for groups acting on simplicial complexes without identifying a fundamental domain. We apply this to the action of the Torelli group on a variant of the complex of curves, yielding an inductive description of the Torelli group in terms of the the subgroups stabilizing simple closed curves on the surface.