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

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.

16 pages, small revisions; to appear in Geom. Dedicata

References in corpus (7)

• Cohomological Aspects of Magnus Expansions
• The free splitting complex of a free group I: Hyperbolicity
• Dimension of the Torelli group for Out(F_n)
• A hyperbolic Out(F_n)-complex
• Relative free splitting and free factor complexes I: Hyperbolicity
• A Birman exact sequence for Aut(F_n)
• On the geometry of a proposed curve complex analogue for $Out(F_n)$

Cited by in corpus (14)

• Generators for the hyperelliptic Torelli group and the kernel of the Burau representation at t=-1
• Integrality in the Steinberg module and the top-dimensional cohomology of SL_n(O_K)
• On finite generation of the Johnson filtrations
• A generating set for the palindromic Torelli group
• On the second homology group of the Torelli subgroup of Aut(F_n)
• Generating the Johnson filtration
• A Birman exact sequence for Aut(F_n)
• The complex of partial bases of a free group
• A finite presentation of the level $2$ principal congruence subgroup of $GL(n;\mathbb{Z})$
• A small normal generating set for the handlebody subgroup of the Torelli group
• Effective finite generation for [IA_n,IA_n] and the Johnson kernel
• Palindromic Automorphisms of Free Nilpotent Groups
• Cutting and Pasting in the Torelli subgroup of Out($F_n$)
• Connectivity of partial basis complexes of freely decomposable groups