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ISSN (electronic): 1364-0380
ISSN (print): 1465-3060
Generating the Johnson filtration

Thomas Church and Andrew Putman

Geometry & Topology 19 (2015) 2217–2255
Abstract

For k 1, let g1(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 1, there exists some Gk 0 such that g1(k) is generated by elements which are supported on subsurfaces whose genus is at most Gk. We also prove similar theorems for the Johnson filtration of Aut(Fn) and for certain mod-p analogues of the Johnson filtrations of both the mapping class group and of Aut(Fn). 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 .

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/