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Finite generation of abelianizations of the genus 3 Johnson kernel and the commutator subgroup of the Torelli group for $\mathrm{Out}(F_3)$

Alexander A. Gaifullin

TL;DR

This work proves that the abelianizations $( K_3^b)^{ ext{ab}}$ and $[ ext{IO}_3, ext{IO}_3]^{ ext{ab}}$ are finitely generated by developing a general criterion for finitely generating modules over Laurent polynomial rings using a subgroup displacement property and $S_{ rak X}$-torsion. The authors extend these ideas to a nilpotency framework, showing $( K_g^{ ext{ab}})$ is nilpotent over a finite-index subgroup $V_g$ with explicit index and index-based bounds, and provide quantitative estimates for genus $3$. They also derive consequences for low-rank second homology, including infinite rank subgroups in $H_2([ ext{IO}_3, ext{IO}_3],Z)$ and conditional relationships with $H_2( I_3^b,Z)$. The methodology blends Johnson-type homomorphisms, SDP, and intricate functional-equation analyses to control abelianizations and homology, yielding a versatile toolkit for finiteness questions in Torelli-type groups. Overall, the paper advances understanding of low-rank Torelli phenomena and offers a robust algebraic framework with potential applications to related automorphism and mapping class group questions.

Abstract

Let $Σ_g^b$ be a compact oriented surface of genus $g$ with $b$ boundary components, where $b\in\{0,1\}$. The Johnson kernel $\mathcal{K}_g^b$ is the subgroup of the mapping class group $\mathrm{Mod}(Σ_g^b)$ generated by Dehn twists about separating simple closed curves. Let $F_n$ be a free group with $n$ generators. The Torelli group for $\mathrm{Out}(F_n)$ is the subgroup $\mathrm{IO}_n\subset\mathrm{Out}(F_n)$ consisting of all outer automorphisms that act trivially on the abelianization of $F_n$. Long standing questions are whether the groups $\mathcal{K}_g^b$ and $[\mathrm{IO}_n,\mathrm{IO}_n]$ or their abelianizations $(\mathcal{K}_g^b)^{\mathrm{ab}}$ and $[\mathrm{IO}_n,\mathrm{IO}_n]^{\mathrm{ab}}$ are finitely generated for $g\ge3$ (respectively, $n\ge3$). During the last 15 years, these questions were answered positively for $g\ge4$ and $n\ge4$, respectively. Nevertheless, the cases of $g=3$ and $n=3$ remained completely unsettled. In this paper, we prove that the abelianizations $(\mathcal{K}_3^b)^{\mathrm{ab}}$ and $[\mathrm{IO}_3,\mathrm{IO}_3]^{\mathrm{ab}}$ are finitely generated. Our approach is based on a new general sufficient condition for a module over a Laurent polynomial ring to be finitely generated as an abelian group.

Finite generation of abelianizations of the genus 3 Johnson kernel and the commutator subgroup of the Torelli group for $\mathrm{Out}(F_3)$

TL;DR

This work proves that the abelianizations and are finitely generated by developing a general criterion for finitely generating modules over Laurent polynomial rings using a subgroup displacement property and -torsion. The authors extend these ideas to a nilpotency framework, showing is nilpotent over a finite-index subgroup with explicit index and index-based bounds, and provide quantitative estimates for genus . They also derive consequences for low-rank second homology, including infinite rank subgroups in and conditional relationships with . The methodology blends Johnson-type homomorphisms, SDP, and intricate functional-equation analyses to control abelianizations and homology, yielding a versatile toolkit for finiteness questions in Torelli-type groups. Overall, the paper advances understanding of low-rank Torelli phenomena and offers a robust algebraic framework with potential applications to related automorphism and mapping class group questions.

Abstract

Let be a compact oriented surface of genus with boundary components, where . The Johnson kernel is the subgroup of the mapping class group generated by Dehn twists about separating simple closed curves. Let be a free group with generators. The Torelli group for is the subgroup consisting of all outer automorphisms that act trivially on the abelianization of . Long standing questions are whether the groups and or their abelianizations and are finitely generated for (respectively, ). During the last 15 years, these questions were answered positively for and , respectively. Nevertheless, the cases of and remained completely unsettled. In this paper, we prove that the abelianizations and are finitely generated. Our approach is based on a new general sufficient condition for a module over a Laurent polynomial ring to be finitely generated as an abelian group.

Paper Structure

This paper contains 28 sections, 64 theorems, 135 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Johnson kernel
  3. Automorphisms of free groups
  4. Method and a general result
  5. Nilpotency property of $\mathcal{K}_g^{\mathrm{ab}}$
  6. Application to the second homology
  7. Organization of the paper
  8. Proof of general sufficient conditon for finite generation
  9. Preliminaries on Johnson homomorphisms
  10. Mapping class groups
  11. Automorphisms of free groups
  12. Subgroup displacement property
  13. $S_{\mathfrak{X}}$-torsion property for the abelianization of a subgroup
  14. Proof of Theorem \ref{['thm_main_K']}
  15. Proof of Theorem \ref{['thm_main_OA']}
  16. ...and 13 more sections

Key Result

Theorem A

Suppose that $g\ge 3$ and $b\in\{0,1\}$. Then $(\mathcal{K}_g^b)^{\mathrm{ab}}$ is a finitely generated abelian group.

Figures (5)

  • Figure 1: Bounding pair map
  • Figure 2: Commutator of a simply intersecting pair
  • Figure 3: Surface $\Sigma_g^b$ and curves $\gamma$, $\alpha_i$, and $\alpha_i'$
  • Figure 4: Surface $\overline{S}\approx\Sigma_0^6$ and curves $\xi_i$, $\eta_i$, and $\zeta_i$
  • Figure 5: Lantern relation

Theorems & Definitions (120)

  • Theorem A
  • Remark 1.1
  • Remark 1.2
  • Theorem B
  • Remark 1.3
  • Remark 1.4
  • Remark 1.5
  • Definition 1.6
  • Definition 1.7
  • Theorem C
  • ...and 110 more