## Geometry & Topology

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

#### Abstract

Let $IAn$ be the Torelli subgroup of $Aut(Fn)$. We give an explicit finite set of generators for $H2(IAn)$ as a $GLn(ℤ)$–module. Corollaries include a version of surjective representation stability for $H2(IAn)$, the vanishing of the $GLn(ℤ)$–coinvariants of $H2(IAn)$, and the vanishing of the second rational homology group of the level $ℓ$ 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
Revised: 8 November 2016
Accepted: 23 December 2016
First available in Project Euclid: 16 November 2017

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

#### 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.