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

Jacob Landgraf

Abstract

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 mapping class groups. Our main results show that these groups are finitely generated, and also that they satisfy an appropriate version of the Birman exact sequence.

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

Abstract

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 mapping class groups. Our main results show that these groups are finitely generated, and also that they satisfy an appropriate version of the Birman exact sequence.

Paper Structure

This paper contains 42 sections, 26 theorems, 75 equations, 15 figures.

Table of Contents

  1. Introduction
  2. Adding boundary components.
  3. Torelli subgroups.
  4. Restriction.
  5. Birman exact sequence.
  6. Finite generation.
  7. Abelianization.
  8. Acknowledgements.
  9. Outline.
  10. Figure conventions.
  11. Preliminaries
  12. Sphere twists.
  13. Action on curves and surfaces.
  14. Sphere twist subgroup.
  15. Restriction Theorem
  16. ...and 27 more sections

Key Result

Theorem A

Let $\iota: M_{n,b} \hookrightarrow M_{m}$ be an embedding, $\iota_*: \mathrm{Out}(F_{n,b}) \to \mathop{\mathrm{Out}}\nolimits(F_m)$ the induced map, and $P$ the induced partition of the boundary components of $M_{n,b}$. Then $IO_{n,b}^{P} = \iota_*^{-1}(IO_m)$.

Figures (15)

  • Figure 1: A copy of $M_{2,2}$ and $M_{1,2}$ glued together to obtain $M_{4}$. We realize $M_{2,2}$ as a 3-sphere with the six indicated open balls removed, then the boundaries of these removed balls are identified according to the arrows (and similarly for $M_{1,2}$). The class $[\alpha]$ need not be fixed by elements of $IO_{2,2}^{}$ with the naïve definition.
  • Figure 2: $M_{5}$ realized by gluing $M_{1,2}$, $M_{1,3}$, and $M_{2,1}$ together along their boundaries as indicated by the arrows.
  • Figure 3: $M_{0,4}$ embedded in $\mathbb{R}^3$.
  • Figure 4: A loop can be surgered into a collection of loops which intersect $\partial M_{n,b}$ exactly twice.
  • Figure 5: The image of $\alpha$ under $\mathop{\mathrm{\widetilde{\mathop{\mathrm{Push}}\nolimits}}}\nolimits(\gamma, T)$ is $\gamma^{-1}\alpha\gamma$. Here, $T$ can be either $T_\partial$ or trivial.
  • ...and 10 more figures

Theorems & Definitions (51)

  • Definition
  • Definition
  • Theorem A: Restriction Theorem
  • Theorem B: Birman exact sequence
  • Remark
  • Theorem 1.1: Magnus
  • Theorem C
  • Theorem D
  • Theorem 1.2: Andreadakis, Bachmuth
  • Theorem E
  • ...and 41 more