On the second homology group of the Torelli subgroup of Aut(Fn)

Day, Matthew¹ ; Putman, Andrew²

¹ Department of Mathematical Sciences, University of Arkansas, Fayetteville, AR, United States
² Department of Mathematics, University of Notre Dame, Notre Dame, IN, United States

Geometry & topology, Tome 21 (2017) no. 5, pp. 2851-2896.

Résumé

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 ${GL}_n\left(\mathbb{Z}\right)$–module. Corollaries include a version of surjective representation stability for $H_2\left({IA}_n\right)$, the vanishing of the ${GL}_n\left(\mathbb{Z}\right)$–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.

Détail

DOI : 10.2140/gt.2017.21.2851

Classification : 20E05, 20E36, 20F05, 20J06
Keywords: automorphism group of free group, Torelli group

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