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

@article{Day2010TheCO, title={The complex of partial bases for Fn and finite generation of the Torelli subgroup of Aut (Fn)}, author={Matthew B. Day and Andrew Putman}, journal={Geometriae Dedicata}, year={2010}, volume={164}, pages={139-153} }

We study the complex of partial bases of a free group, which is an analogue for Aut(Fn) 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(Fn) 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(Fn) is finitely generated.

## 21 Citations

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

- Mathematics
- 2017

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…

Cutting and Pasting in the Torelli subgroup of Out($F_n$)

- Mathematics
- 2021

Using ideas from 3-manifolds, Hatcher–Wahl defined a notion of automorphism groups of free groups with boundary. We study their Torelli subgroups, adapting ideas introduced by Putman for surface…

On finite generation of the Johnson filtrations

- MathematicsJournal of the European Mathematical Society
- 2021

We prove that every term of the lower central series and Johnson filtrations of the Torelli subgroups of the mapping class group and the automorphism group of a free group is finitely generated in a…

The complex of partial bases of a free group

- Mathematics
- 2017

We prove that the simplicial complex whose simplices are the nonempty partial bases of $\mathbb{F}_n$ is homotopy equivalent to a wedge of $(n-1)$-spheres. Moreover, we show that it is Cohen-Macaulay.

A small normal generating set for the handlebody subgroup of the Torelli group

- MathematicsGeometriae Dedicata
- 2018

We prove that the handlebody subgroup of the Torelli group of an orientable surface is generated by genus one BP-maps . As an application, we give a normal generating set for the handlebody subgroup…

Generators for the hyperelliptic Torelli group and the kernel of the Burau representation at $$t=-1$$t=-1

- Mathematics
- 2015

We prove that the hyperelliptic Torelli group is generated by Dehn twists about separating curves that are preserved by the hyperelliptic involution. This verifies a conjecture of Hain. The…

N ov 2 01 7 On finite generation of the Johnson filtrations

- Mathematics
- 2017

We prove that every term of the lower central series and Johnson filtrations of the Torelli subgroups of the mapping class group and the automorphism group of a free group is finitely generated in a…

The complex of partial bases of a free group

- Mathematics
- 2020

We prove that the simplicial complex whose simplices are the nonempty partial bases of Fn is homotopy equivalent to a wedge of (n−1) ‐spheres. Moreover, we show that it is Cohen–Macaulay.

Generating the Johnson filtration

- Mathematics
- 2013

For k >= 1, let Torelli_g^1(k) be the k-th term in the Johnson filtration of the mapping class group of a genus g surface with one boundary component. We prove that for all k, there exists some G_k…

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