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Geometry of extensions of free groups via automorphisms with fixed points on the complex of free factors

Pritam Ghosh, Funda Gültepe

TL;DR

This work develops a comprehensive framework for understanding when free-by-free extensions $E_Q$ arising from subgroups $Q<\mathrm{Out}(\mathbb{F})$ are hyperbolic or relatively hyperbolic, by marrying dynamics on the free factor complex $\mathcal{FF}$ with train-track laminations. Central to the analysis is allowing fixed points on $\mathcal{FF}$ for outer automorphisms and quantifying their separation via the notions of sufficiently different automorphisms and non-attracting sinks $\mathcal{K}^*_\phi$, together with weak attraction theory and CT maps. The paper proves that hyperbolicity of $E_Q$ is equivalent to the generators being atoroidal with pairwise disjoint laminations, and more generally characterizes when extensions are hyperbolic relative to cusps based on sinks and the geometry of laminations, including cases involving partial and relative fully irreducibles. By developing small displacement set analysis and subfactor projections, the authors derive a suite of criteria that unify the hyperbolic and relatively hyperbolic behavior of free-by-free extensions under fixed-point dynamics on $\mathcal{FF}$. The results extend and complement prior work on convex cocompactness and mapping class group analogies, and provide concrete mechanisms to construct hyperbolic or relatively hyperbolic extensions via high powers of suitably chosen automorphisms.

Abstract

We give conditions of an extension of a free group to be hyperbolic and relatively hyperbolic using the dynamics of the action of $\out$ on the complex of free factors combined with the weak attraction theory. We work with subgroups of exponentially growing outer automorphisms and instead of using a standard pingpong argument with loxodromics, we allow fixed points for the action and investigate the geometry of the extension group when the fixed points of the automorphisms on the complex of free factors are sufficiently far apart.

Geometry of extensions of free groups via automorphisms with fixed points on the complex of free factors

TL;DR

This work develops a comprehensive framework for understanding when free-by-free extensions arising from subgroups are hyperbolic or relatively hyperbolic, by marrying dynamics on the free factor complex with train-track laminations. Central to the analysis is allowing fixed points on for outer automorphisms and quantifying their separation via the notions of sufficiently different automorphisms and non-attracting sinks , together with weak attraction theory and CT maps. The paper proves that hyperbolicity of is equivalent to the generators being atoroidal with pairwise disjoint laminations, and more generally characterizes when extensions are hyperbolic relative to cusps based on sinks and the geometry of laminations, including cases involving partial and relative fully irreducibles. By developing small displacement set analysis and subfactor projections, the authors derive a suite of criteria that unify the hyperbolic and relatively hyperbolic behavior of free-by-free extensions under fixed-point dynamics on . The results extend and complement prior work on convex cocompactness and mapping class group analogies, and provide concrete mechanisms to construct hyperbolic or relatively hyperbolic extensions via high powers of suitably chosen automorphisms.

Abstract

We give conditions of an extension of a free group to be hyperbolic and relatively hyperbolic using the dynamics of the action of on the complex of free factors combined with the weak attraction theory. We work with subgroups of exponentially growing outer automorphisms and instead of using a standard pingpong argument with loxodromics, we allow fixed points for the action and investigate the geometry of the extension group when the fixed points of the automorphisms on the complex of free factors are sufficiently far apart.
Paper Structure (24 sections, 28 theorems, 10 equations)

This paper contains 24 sections, 28 theorems, 10 equations.

Table of Contents

  1. Introduction
  2. Fixed points on $\mathcal{F}\mathcal{F}$ and hyperbolicity of the extension
  3. Sufficiently different automorphisms, non-attracting sink and the geometry of the extension group
  4. Plan of the paper:
  5. Acknowledgements
  6. Marked graphs, circuits and path:
  7. Topological representative, EG strata, NEG strata:
  8. Weak topology
  9. Free factor systems and malnormal subgroup systems.
  10. Attracting laminations and nonattracting subgroup systems:
  11. Completely split improved relative train track (CT) maps
  12. Critical Constant
  13. Admissible subgroup systems and relative hyperbolicity
  14. Detecting Obstructions to (relative) hyperbolicity on the complex of free factors
  15. Sufficiently different outer automorphisms and their dynamics on $\mathcal{F}\mathcal{F}$
  16. ...and 9 more sections

Key Result

Theorem 1.1

Let $\phi_1, \ldots , \phi_k$ be a collection of exponentially growing outer automorphisms such that no pair of automorphisms have a common power. Then the following are equivalent:

Theorems & Definitions (59)

  • Theorem 1.1
  • Theorem
  • Theorem 1.2
  • Theorem 1.3
  • Lemma 2.1
  • Lemma 2.2: BFH-00, Section 2.6
  • Lemma 2.3
  • Lemma 2.4
  • Definition 2.5
  • Definition 2.6
  • ...and 49 more