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

TL;DR

This paper provides an explicit finite generating set for the second homology group of the Torelli subgroup of automorphisms of free groups, revealing stability properties and vanishing results relevant to algebraic topology and group theory.

Contribution

It introduces a new recursive group presentation for IA_n and derives a finite generating set for H_2(IA_n) as a GL_n(Z)-module, advancing understanding of its structure.

Findings

Finite generating set for H_2(IA_n) as a GL_n(Z)-module

Surjective representation stability for H_2(IA_n)

Vanishing of GL_n(Z)-coinvariants of H_2(IA_n) and second rational homology

Abstract

Let IA_n be the Torelli subgroup of Aut(F_n). We give an explicit finite set of generators for H_2(IA_n) as a GL_n(Z)-module. Corollaries include a version of surjective representation stability for H_2(IA_n), the vanishing of the GL_n(Z)-coinvariants of H_2(IA_n), and the vanishing of the second rational homology group of the level l congruence subgroup of Aut(F_n). Our generating set is derived from a new group presentation for IA_n which is infinite but which has a simple recursive form.