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

Abstract

Let $I{A}_n$ denote the group of $IA$-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 $I{A}_n$ 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 $k^{\mathrm{th}}$ terms of the lower central series ${\gamma }_{k}I{A}_n$ and ${\gamma }_k{\mathcal{I}}_n^b$ 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 ${\gamma }_{2}I{A}_n=[I{A}_n,I{A}_n]$ 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.

arXiv comment37 pages, 4 figures; final version

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