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An Infinite Presentation of the Torelli Group

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Abstract

In this paper, we construct an infinite 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 all simple closed curves.

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  1. Department of Mathematics, MIT, 2-306, 77 Massachusetts Avenue, Cambridge, MA, 02139-4307, USA

    Andrew Putman

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  1. Andrew Putman
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Putman, A. An Infinite Presentation of the Torelli Group. Geom. Funct. Anal. 19, 591–643 (2009). https://doi.org/10.1007/s00039-009-0006-6

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