On the second homology group of the Torelli subgroup of $\mathrm{Aut}(F_n)$

Abstract

Let ${IA}_n$ be the Torelli subgroup of $Aut\left(F_n\right)$. We give an explicit finite set of generators for $H_2\left({IA}_n\right)$ as a ${GL}_n\left(\mathbb{Z}\right)$–module. Corollaries include a version of surjective representation stability for $H_2\left({IA}_n\right)$, the vanishing of the ${GL}_n\left(\mathbb{Z}\right)$–coinvariants of $H_2\left({IA}_n\right)$, and the vanishing of the second rational homology group of the level $\ell$ congruence subgroup of $Aut\left(F_n\right)$. Our generating set is derived from a new group presentation for ${IA}_n$ which is infinite but which has a simple recursive form.