Generating the Johnson filtration

Church, Thomas; Putman, Andrew

doi:10.2140/gt.2015.19.2217

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Generating the Johnson filtration

Thomas Church and Andrew Putman

Geometry & Topology 19 (2015) 2217–2255

DOI: 10.2140/gt.2015.19.2217

Abstract

For $k \geq 1$ , let $ℐ_{g}^{1} (k)$ be the $k^{th}$ term in the Johnson filtration of the mapping class group of a genus $g$ surface with one boundary component. We prove that for all $k \geq 1$ , there exists some $G_{k} \geq 0$ such that $ℐ_{g}^{1} (k)$ 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 $Aut (F_{n})$ and for certain mod- $p$ analogues of the Johnson filtrations of both the mapping class group and of $Aut (F_{n})$ . 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

References

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/

Volume 19, issue 4 (2015)