On finiteness properties of the Johnson filtrations

Abstract

Let $\Gamma$ be either the automorphism group of the free group of rank $n\ge 4$ or the mapping class group of an orientable surface of genus $n\ge 12$ with at most $1$ boundary component, and let $G$ be either the subgroup of $\mathrm{IA}$-automorphisms or the Torelli subgroup of $\Gamma$. For $N\in \mathbb{N}$ denote by ${\gamma }_{N}G$ the $N$th term of the lower central series of $G$. We prove that

(i) any subgroup of $G$ containing ${\gamma }_{2}G=[G,G]$ (in particular, the Johnson kernel in the mapping class group case) is finitely generated;

(ii) if $N=2$ or $n\ge 8N-4$ and $K$ is any subgroup of $G$ containing ${\gamma }_{N}G$ (for instance, $K$ can be the $N$th term of the Johnson filtration of $G$), then $G/[K,K]$ is nilpotent and hence the Abelianization of $K$ is finitely generated;

(iii) if $H$ is any finite-index subgroup of $\Gamma$ containing ${\gamma }_{N}G$, with $N$ as in (ii), then $H$ has finite Abelianization.