Effective finite generation for [IAn, IAn] and the Johnson kernel

Let IA(n) denote the group of IA-automorphisms of a free group of rank n, and let I-n(b) denote the Torelli subgroup of the mapping class group of an orientable surface of genus n with b boundary components, b = 0, 1. In 1935, Magnus proved that IA(n) is finitely generated for all n, and in 1983, Johnson proved that I-n(b) is finitely generated for n >= 3. It was recently shown that for each k is an element of N, the k-th terms of the lower central series gamma(k)IA(n) and gamma I-k(n)b are finitely generated when n >> k; however, no information about finite generating sets was known for k > 1. The main goal of this paper is to construct an explicit finite generating set for gamma(2)IA(n) = [IA(n), IA(n)] and almost explicit finite generating sets for gamma I-2(n)b and the Johnson kernel, which contains gamma I-2(n)b as a finite index subgroup.