Effective finite generation for $[\mathrm{ IA}_n,\mathrm{ IA}_n]$ and the Johnson kernel

Abstract

Let \mathrm{ IA}_n denote the group of \mathrm{ IA} -automorphisms of a free group of rank n , and let \mathcal 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 \mathrm{ IA}_n is finitely generated for all n , and in 1983, Johnson proved that \mathcal I_n^b is finitely generated for n\geq 3 . It was recently shown that for each k\in{\mathbb N} , the k -th terms of the lower central series \gamma_k \mathrm{ IA}_n and \gamma_k\mathcal I_n^b are finitely generated when n\gg 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 \mathrm{ IA}_n = [\mathrm{ IA}_n,\mathrm{ IA}_n] and almost explicit finite generating sets for \gamma_2\mathcal I_n^b and the Johnson kernel, which contains \gamma_2\mathcal I_n^b as a finite index subgroup.