#### Volume 19, issue 4 (2015)

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Generating the Johnson filtration

### Thomas Church and Andrew Putman

Geometry & Topology 19 (2015) 2217–2255
##### Abstract

For $k\ge \mathfrak{1}$, let ${\mathsc{ℐ}}_{g}^{\mathfrak{1}}\left(k\right)$ be the 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 ${\mathsc{ℐ}}_{g}^{\mathfrak{1}}\left(k\right)$ 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-$\phantom{\rule{0.3em}{0ex}}p$ 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 $ℤ$.

##### Keywords
Mapping class group, Torelli group, Johnson filtration, automorphism group of free group, FI–modules
##### Mathematical Subject Classification 2010
Primary: 20F05, 57S05
Secondary: 57M07, 57N05
##### Publication
Received: 23 December 2013
Revised: 21 August 2014
Accepted: 3 October 2014
Published: 29 July 2015
Proposed: Shigeyuki Morita
Seconded: Danny Calegari, Cameron Gordon
##### Authors
 Thomas Church Department of Mathematics Stanford University 450 Serra Mall Stanford, CA 94305 USA http://math.stanford.edu/~church Andrew Putman Department of Mathematics Rice University MS 136 6100 Main St. Houston, TX 77005 http://www.math.rice.edu/~andyp/