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Hyperbolic actions and 2nd bounded cohomology of subgroups of $\mathsf{Out}(F_n)$. Part II: Finite lamination subgroups

Michael Handel, Lee Mosher

TL;DR

The paper advances the $H^2_b$-alternative for finitely generated subgroups of $\mathsf{Out}(F_n)$ by analyzing finite lamination subgroups. It constructs automorphic lifts and hyperbolic actions via a two-pronged approach: a one-edge case and a multi-edge case, employing flaring arguments, relative train track technology, and the Mj–Sardar combination theorem. Central to the work is the suspension space $\mathcal{S}$ built from a cone-off tree $T^*$ and an abelian disintegration framework that yields nonelementary, WWPD-rich actions on hyperbolic spaces. The results yield that, for non-virtually-abelian finite lamination subgroups with abelian restrictions, a finite-index subgroup acts on a hyperbolic space with elliptic/loxodromic elements, nonelementarity, and WWPD elements in the commutator, thereby forcing uncountable $H^2_b$. The construction integrates deep Out$(F_n)$-techniques (CTs, disintegration, principal automorphisms) with geometric group theory tools (hyperbolicity, flaring, and combination theorems) to certify nontrivial second bounded cohomology. The findings extend the scope of bounded cohomology methods in free-group automorphism groups and illuminate the dynamics of finite lamination subgroups under hyperbolic actions.

Abstract

This is the second part of a two part work in which we prove that for every finitely generated subgroup $Γ< \mathsf{Out}(F_n)$, either $Γ$ is virtually abelian or its second bounded cohomology $H^2_b(Γ;\mathbb{R})$ contains an embedding of $\ell^1$. Here in Part II we focus on finite lamination subgroups $Γ$ --- meaning that the set of all attracting laminations of elements of $Γ$ is finite --- and on the construction of hyperbolic actions of those subgroups to which the general theory of Part I is applicable.

Hyperbolic actions and 2nd bounded cohomology of subgroups of $\mathsf{Out}(F_n)$. Part II: Finite lamination subgroups

TL;DR

The paper advances the -alternative for finitely generated subgroups of by analyzing finite lamination subgroups. It constructs automorphic lifts and hyperbolic actions via a two-pronged approach: a one-edge case and a multi-edge case, employing flaring arguments, relative train track technology, and the Mj–Sardar combination theorem. Central to the work is the suspension space built from a cone-off tree and an abelian disintegration framework that yields nonelementary, WWPD-rich actions on hyperbolic spaces. The results yield that, for non-virtually-abelian finite lamination subgroups with abelian restrictions, a finite-index subgroup acts on a hyperbolic space with elliptic/loxodromic elements, nonelementarity, and WWPD elements in the commutator, thereby forcing uncountable . The construction integrates deep Out-techniques (CTs, disintegration, principal automorphisms) with geometric group theory tools (hyperbolicity, flaring, and combination theorems) to certify nontrivial second bounded cohomology. The findings extend the scope of bounded cohomology methods in free-group automorphism groups and illuminate the dynamics of finite lamination subgroups under hyperbolic actions.

Abstract

This is the second part of a two part work in which we prove that for every finitely generated subgroup , either is virtually abelian or its second bounded cohomology contains an embedding of . Here in Part II we focus on finite lamination subgroups --- meaning that the set of all attracting laminations of elements of is finite --- and on the construction of hyperbolic actions of those subgroups to which the general theory of Part I is applicable.

Paper Structure

This paper contains 84 sections, 33 theorems, 138 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Remarks: WPD versus WWPD.
  3. Prerequisites from the theory of $\mathop{\mathrm{\mathsf{Out}}}\nolimits(F_n)$.
  4. Lifting to an automorphism group
  5. Definition of automorphic lifts
  6. A sufficient condition to be abelian.
  7. Constructing automorphic lifts: proof of Proposition \ref{['LemmaAutLiftExists']}
  8. Proof that the Hyperbolic Action Theorem implies Theorem C
  9. Automorphic extensions of free splitting actions
  10. Free splittings.
  11. Twisted equivariance (functional notation).
  12. Remark.
  13. Free splittings of co-edge number $1$.
  14. The stabilizer of a free splitting and its automorphic extension
  15. Remark:
  16. ...and 69 more sections

Key Result

Proposition 2.3

If $\Gamma < \mathop{\mathrm{IA}}\nolimits_n({\mathbb Z}/3)$ is a finitely generated, finite lamination subgroup which is not (virtually) abelian but which has (virtually) abelian restrictions, then there exists an automorphic lift $\Gamma \mapsto \mathop{\mathrm{\mathsf{Aut}}}\nolimits(A)$.

Figures (2)

  • Figure 1: Notation associated to an automorphic lift $\rho : \Gamma \to \mathop{\mathrm{\mathsf{Aut}}}\nolimits(A)$ with image $\widehat{{\cal H}}$. The group $\widehat{{\cal H}}$ is not virtually abelian, the quotient ${\cal H}$ is virtually abelian, and the kernel $J$ is free of rank $\ge 2$ (possibly infinite). The horizontal rows are exact.
  • Figure 2: The group $\widehat{{\cal H}}$ and associated objects in a commutative diagram with short exact rows. The automorphism $\Phi \in \widehat{{\cal H}} \in \mathop{\mathrm{\mathsf{Aut}}}\nolimits(F_n)$ is a chosen pre-image of $\phi \in {\cal H}$ and hence satisfies $\widehat{\omega}_u(\Phi)=\omega_u(\phi)=1$.

Theorems & Definitions (63)

  • Definition 2.1: (Virtually) Abelian restrictions
  • Definition 2.2
  • Proposition 2.3
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • proof
  • Lemma 2.7
  • Lemma 2.8
  • proof
  • ...and 53 more