On finiteness properties of the Johnson filtrations

Let $A$ denote either the automorphism group of the free group of rank $n$ or the mapping class group of an orientable surface of genus $n$ with at most 1 boundary component, and let $G$ be either the subgroup of IA-automorphisms or the Torelli subgroup of $A$, respectively. I will discuss various finiteness properties of subgroups containing $G_N$, the $N$-th term of the lower central series of $G$, for sufficiently small $N$. In particular, I will explain why
(1) If $n \geq 4N-1$, then any subgroup of G containing $G_N$ (e.g. the $N$-th term of the Johnson filtration) is finitely generated
(2) If $n \geq 8N-3$, then any finite index subgroup of $A$ containing $G_N$ has finite abelianization.
The talk will be based on a joint work with Sue He and a joint work with Tom Church and Andrew Putman