Generating the Johnson filtration

Abstract

For $k\ge \mathfrak{1}$, let ${\mathcal{ℐ}}_g^{\mathfrak{1}}\left(k\right)$ be the $k^{\text{ th}}$ term in the Johnson filtration of the mapping class group of a genus $g$ surface with one boundary component. We prove that for all $k\ge \mathfrak{1}$, there exists some $G_k\ge \mathfrak{0}$ such that ${\mathcal{ℐ}}_g^{\mathfrak{1}}\left(k\right)$ is generated by elements which are supported on subsurfaces whose genus is at most $G_k$. We also prove similar theorems for the Johnson filtration of $Aut\left(F_n\right)$ and for certain mod-$p$ analogues of the Johnson filtrations of both the mapping class group and of $Aut\left(F_n\right)$. The main tools used in the proofs are the related theories of FI–modules (due to the first author with Ellenberg and Farb) and central stability (due to the second author), both of which concern the representation theory of the symmetric groups over $\mathbb{Z}$.