The complex of partial bases for F_n and finite generation of the Torelli subgroup of Aut(F_n)

TL;DR

This paper investigates the complex of partial bases for free groups, proving its connectivity properties and using these topological results to provide a new proof that the Torelli subgroup of automorphisms of free groups is finitely generated.

Contribution

It introduces a topological approach to study the complex of partial bases and establishes its connectivity, leading to a novel proof of finite generation of the Torelli subgroup.

Findings

The complex of partial bases is connected and simply connected.

The quotient of this complex by the Torelli subgroup is highly connected.

Provides a new topological proof of finite generation of the Torelli subgroup.

Abstract

We study the complex of partial bases of a free group, which is an analogue for $\Aut(F_n)$ of the curve complex for the mapping class group. We prove that it is connected and simply connected, and we also prove that its quotient by the Torelli subgroup of $\Aut(F_n)$ is highly connected. Using these results, we give a new, topological proof of a theorem of Magnus that asserts that the Torelli subgroup of $\Aut(F_n)$ is finitely generated.