Geometry & Topology

On the second homology group of the Torelli subgroup of $\mathrm{Aut}(F_n)$

Matthew Day and Andrew Putman

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Abstract

Let IAn be the Torelli subgroup of Aut(Fn). We give an explicit finite set of generators for H2(IAn) as a GLn()–module. Corollaries include a version of surjective representation stability for H2(IAn), the vanishing of the GLn()–coinvariants of H2(IAn), and the vanishing of the second rational homology group of the level congruence subgroup of Aut(Fn). Our generating set is derived from a new group presentation for IAn which is infinite but which has a simple recursive form.

Article information

Source
Geom. Topol., Volume 21, Number 5 (2017), 2851-2896.

Dates
Received: 22 October 2015
Revised: 8 November 2016
Accepted: 23 December 2016
First available in Project Euclid: 16 November 2017

Permanent link to this document
https://projecteuclid.org/euclid.gt/1510859276

Digital Object Identifier
doi:10.2140/gt.2017.21.2851

Mathematical Reviews number (MathSciNet)
MR3687109

Zentralblatt MATH identifier
06774935

Subjects
Primary: 20E05: Free nonabelian groups 20E36: Automorphisms of infinite groups [For automorphisms of finite groups, see 20D45] 20F05: Generators, relations, and presentations 20J06: Cohomology of groups

Keywords
automorphism group of free group Torelli group

Citation

Day, Matthew; Putman, Andrew. On the second homology group of the Torelli subgroup of $\mathrm{Aut}(F_n)$. Geom. Topol. 21 (2017), no. 5, 2851--2896. doi:10.2140/gt.2017.21.2851. https://projecteuclid.org/euclid.gt/1510859276


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Supplemental materials

  • GAP code for the calculations.