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 $n\ge 3$. 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.