An infinite presentation of the Torelli group

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.

52 pages, 14 figures, 1 table, very heavily revised and expanded, section on obtaining presentations from group actions spun off as separate paper; to appear in GAFA

References in corpus (1)

• Cutting and Pasting in the Torelli Group

Cited by in corpus (12)

• Generators for the hyperelliptic Torelli group and the kernel of the Burau representation at t=-1
• The Picard group of the moduli space of curves with level structures
• The complex of partial bases for F_n and finite generation of the Torelli subgroup of Aut(F_n)
• The second rational homology group of the moduli space of curves with level structures
• On the second homology group of the Torelli subgroup of Aut(F_n)
• Positive factorizations of mapping classes
• Obtaining presentations from group actions without making choices
• Small generating sets for the Torelli group
• Equivariant group presentations and the second homology group of the Torelli group
• Surface groups, infinite generating sets, and stable commutator length
• Apartment classes of integral symplectic groups
• Connectivity of partial basis complexes of freely decomposable groups