Geometry & Topology

Generating the Johnson filtration

Thomas Church and Andrew Putman

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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 .

Article information

Geom. Topol., Volume 19, Number 4 (2015), 2217-2255.

Received: 23 December 2013
Revised: 21 August 2014
Accepted: 3 October 2014
First available in Project Euclid: 16 November 2017

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 20F05: Generators, relations, and presentations 57S05: Topological properties of groups of homeomorphisms or diffeomorphisms
Secondary: 57M07: Topological methods in group theory 57N05: Topology of $E^2$ , 2-manifolds

Mapping class group Torelli group Johnson filtration automorphism group of free group FI–modules


Church, Thomas; Putman, Andrew. Generating the Johnson filtration. Geom. Topol. 19 (2015), no. 4, 2217--2255. doi:10.2140/gt.2015.19.2217.

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