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The geometry of subgroups of mapping tori of free groups

Marco Linton

TL;DR

This work shows that any finitely generated mapping torus $M(\psi)$ of a free group $\mathbb{F}$ admits a canonical finite collection $\mathcal{P}$ of maximal sub-mapping tori such that $(M(\psi),\mathcal{P})$ is relatively hyperbolic and locally relatively quasi-convex. It develops a criterion for relative quasi-convexity in graphs of relatively hyperbolic spaces via the Mj–Reeves combination theorem and Feighn–Handel graph-pair techniques, enabling a detailed analysis of finitely generated subgroups and upgrades of Dehn functions, the conjugacy problem, and the finitely generated intersection property to arbitrary finitely generated free-group mapping tori. The paper characterizes when hyperbolic mapping tori are locally quasi-convex in terms of primitivity rank $\pi(w)$ for one-relator groups, yielding effective coherence and LERF in many instances, and connects these results to virtually free-by-cyclic groups and 3-manifold analogues. Overall, it extends the structural and algorithmic understanding of free-by-cyclic groups from the finitely generated base case to arbitrary finitely generated free groups, providing tools and consequences for subgroup dynamics and decision problems.

Abstract

We show that finitely generated mapping tori of free groups have a canonical collection of maximal sub-mapping tori of finitely generated free groups with respect to which they are relatively hyperbolic and locally relatively quasi-convex. As a consequence, we characterise locally quasi-convex hyperbolic groups amongst free-by-cyclic and one-relator groups. We also upgrade several known results for mapping tori of finitely generated free groups to the general case, such as the computations of Dehn functions, the solution to the conjugacy problem and the characterisation of the finitely generated intersection property.

The geometry of subgroups of mapping tori of free groups

TL;DR

This work shows that any finitely generated mapping torus of a free group admits a canonical finite collection of maximal sub-mapping tori such that is relatively hyperbolic and locally relatively quasi-convex. It develops a criterion for relative quasi-convexity in graphs of relatively hyperbolic spaces via the Mj–Reeves combination theorem and Feighn–Handel graph-pair techniques, enabling a detailed analysis of finitely generated subgroups and upgrades of Dehn functions, the conjugacy problem, and the finitely generated intersection property to arbitrary finitely generated free-group mapping tori. The paper characterizes when hyperbolic mapping tori are locally quasi-convex in terms of primitivity rank for one-relator groups, yielding effective coherence and LERF in many instances, and connects these results to virtually free-by-cyclic groups and 3-manifold analogues. Overall, it extends the structural and algorithmic understanding of free-by-cyclic groups from the finitely generated base case to arbitrary finitely generated free groups, providing tools and consequences for subgroup dynamics and decision problems.

Abstract

We show that finitely generated mapping tori of free groups have a canonical collection of maximal sub-mapping tori of finitely generated free groups with respect to which they are relatively hyperbolic and locally relatively quasi-convex. As a consequence, we characterise locally quasi-convex hyperbolic groups amongst free-by-cyclic and one-relator groups. We also upgrade several known results for mapping tori of finitely generated free groups to the general case, such as the computations of Dehn functions, the solution to the conjugacy problem and the characterisation of the finitely generated intersection property.

Paper Structure

This paper contains 37 sections, 54 theorems, 101 equations.

Table of Contents

  1. Introduction
  2. Quasi-convex subgroups of mapping tori
  3. One-relator groups
  4. Summary of the article
  5. Preliminaries
  6. Graphs and graph maps
  7. Folds and subgroups of free groups
  8. Free factor systems
  9. Relatively hyperbolic spaces
  10. Relatively hyperbolic groups and quasi-convex subgroups
  11. Criteria for relative hyperbolicity and quasi-convexity
  12. The combination theorem for trees of relatively hyperbolic spaces
  13. Relatively quasi-convex vertex spaces
  14. Quasi-geodesics in a tree of hyperbolic spaces
  15. Relatively quasi-convex subgroups
  16. ...and 22 more sections

Key Result

Theorem 1.1

Let $\mathbb{F}$ be a free group, $\psi\colon \mathbb{F}\to \mathbb{F}$ a monomorphism and let $G \leqslant M(\psi)$ be a finitely generated non-free subgroup of the mapping torus. There is a (possibly empty) canonical finite collection of (conjugacy classes of) subgroups $\mathcal{P}$ of $G$, each

Theorems & Definitions (113)

  • Theorem 1.1
  • Corollary 1.2
  • Conjecture 1.3
  • Theorem 1.4
  • Corollary 1.5
  • Corollary 1.6
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • ...and 103 more