Volume 21, issue 5 (2017)

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

Matthew Day and Andrew Putman

Geometry & Topology 21 (2017) 2851–2896
Abstract

Let ${IA}_{n}$ be the Torelli subgroup of $Aut\left({F}_{n}\right)$. We give an explicit finite set of generators for ${H}_{2}\left({IA}_{n}\right)$ as a ${GL}_{n}\left(ℤ\right)$–module. Corollaries include a version of surjective representation stability for ${H}_{2}\left({IA}_{n}\right)$, the vanishing of the ${GL}_{n}\left(ℤ\right)$–coinvariants of ${H}_{2}\left({IA}_{n}\right)$, and the vanishing of the second rational homology group of the level $\ell$ congruence subgroup of $Aut\left({F}_{n}\right)$. Our generating set is derived from a new group presentation for ${IA}_{n}$ which is infinite but which has a simple recursive form.

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Keywords
automorphism group of free group, Torelli group
Mathematical Subject Classification 2010
Primary: 20E05, 20E36, 20F05, 20J06
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