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

TL;DR

This paper constructs explicit finite generating sets for the second terms of the lower central series of certain automorphism and mapping class groups, advancing understanding of their algebraic structure.

Contribution

It provides the first explicit finite generating sets for nd terms of these groups, including the Johnson kernel, which was previously unknown.

Findings

Explicit finite generating set for IA_n

Almost explicit generating sets for  and the Johnson kernel

Advances understanding of the algebraic structure of these groups

Abstract

Let $IA_n$ 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 $IA_n$ 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 $k^{\rm th}$ terms of the lower central series $\gamma_k IA_n$ 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 $\gamma_2 IA_n = [IA_n,IA_n]$ 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.