Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Pseudo-Anosov Flows: A Plane Approach

Thomas Barthelmé, Kathryn Mann

TL;DR

This work develops a plane-centered, topological framework for pseudo-Anosov flows on closed 3-manifolds by focusing on the orbit space and Anosov-like group actions on bifoliated planes. It proves that the induced action of the fundamental group on the orbit space determines the flow up to unoriented orbit equivalence (Barbot-type results) and builds a robust theory of Anosov-like actions, including a trichotomy (trivial, skew, or branching with prongs/non-Hausdorff leaf spaces), density results, and a dynamic-combinatorial toolkit based on lozenges, perfect fits, and scalloped regions. The paper derives structural consequences such as exponential growth of the fundamental group via boundary dynamics, rigidity results for abelian normal subgroups, and a generating-set description for Anosov-like groups. With a closing lemma translated into orbit-space language, it connects local orbit dynamics to global algebraic structure, enabling classifications and rigidity phenomena for flows on 3-manifolds and their orbit spaces. Overall, this approach unifies 1- and 2-dimensional dynamics through the orbit space, yielding precise, computable portraits of flow dynamics and their topological implications for 3-manifold geometry.

Abstract

In this text we (re)-tell the theory of pseudo-Anosov flows on 3-manifolds with the orbit space as the central character; via a streamlined framework called {\em Anosov-like group actions}. This brings a simplified and unified perspective, and allows for the import of tools from 1- and 2-dimensional topological dynamics. In so doing, we are able to give a relatively efficient presentation (or re-presentation) of foundational results in the field which appeared over several decades and many papers. We hope that the reader, whether a beginning or more advanced topologist or dynamicist, finds this a welcoming invitation to the rich theory of pseudo-Anosov flows.

Pseudo-Anosov Flows: A Plane Approach

TL;DR

This work develops a plane-centered, topological framework for pseudo-Anosov flows on closed 3-manifolds by focusing on the orbit space and Anosov-like group actions on bifoliated planes. It proves that the induced action of the fundamental group on the orbit space determines the flow up to unoriented orbit equivalence (Barbot-type results) and builds a robust theory of Anosov-like actions, including a trichotomy (trivial, skew, or branching with prongs/non-Hausdorff leaf spaces), density results, and a dynamic-combinatorial toolkit based on lozenges, perfect fits, and scalloped regions. The paper derives structural consequences such as exponential growth of the fundamental group via boundary dynamics, rigidity results for abelian normal subgroups, and a generating-set description for Anosov-like groups. With a closing lemma translated into orbit-space language, it connects local orbit dynamics to global algebraic structure, enabling classifications and rigidity phenomena for flows on 3-manifolds and their orbit spaces. Overall, this approach unifies 1- and 2-dimensional dynamics through the orbit space, yielding precise, computable portraits of flow dynamics and their topological implications for 3-manifold geometry.

Abstract

In this text we (re)-tell the theory of pseudo-Anosov flows on 3-manifolds with the orbit space as the central character; via a streamlined framework called {\em Anosov-like group actions}. This brings a simplified and unified perspective, and allows for the import of tools from 1- and 2-dimensional topological dynamics. In so doing, we are able to give a relatively efficient presentation (or re-presentation) of foundational results in the field which appeared over several decades and many papers. We hope that the reader, whether a beginning or more advanced topologist or dynamicist, finds this a welcoming invitation to the rich theory of pseudo-Anosov flows.

Paper Structure

This paper contains 71 sections, 145 theorems, 20 equations, 68 figures.

Table of Contents

  1. From pseudo-Anosov flows to the orbit space
  2. Definitions
  3. Anosov and topological Anosov flows
  4. Pseudo-Anosov flows
  5. Expansive flows
  6. Examples
  7. Suspension flows
  8. Geodesic flow
  9. The examples of Handel and Thurston
  10. The DA-construction and Franks--Williams example
  11. Dehn--Fried surgery
  12. The Bonatti--Langevin example
  13. A cautionary non-example
  14. The orbit space of a pseudo-Anosov flow
  15. Topology of stable, unstable, and orbit foliations in $\widetilde{M}$
  16. ...and 56 more sections

Key Result

Theorem 1.1.4

Let $\varphi$ be an Anosov flow. There exists a $C^0$ neighborhood of $\varphi$ in the space of maps $M \times \mathbb R \to M$ such that any Anosov flow in this neighborhood is orbit equivalent to $\varphi$. Furthermore, if $\psi$ is any $C^1$ flow that is sufficiently close to $\varphi$ in the $C^

Figures (68)

  • Figure 1.1: A 3-dimensional 3-prong singularity.
  • Figure 1.2: If $x, y$ are sufficiently nearby but not on the same local sheet of $\mathcal{F}^{s}$, then $x$ fellow-travels a point $z$ which does not fellow-travel $y$.
  • Figure 1.3:
  • Figure 1.4: The DA bifurcation
  • Figure 1.5: A $(1,1)$-Dehn twist
  • ...and 63 more figures

Theorems & Definitions (344)

  • Definition 1.1.1
  • Definition 1.1.2
  • Remark 1.1.3
  • Theorem 1.1.4: Structural stability Ano67
  • Proposition 1.1.5: Anosov Ano63
  • Proposition 1.1.6: See, e.g., FH19 § 6.1
  • Definition 1.1.7
  • Remark 1.1.8: Topological flows and time change
  • Definition 1.1.9
  • Definition 1.1.10
  • ...and 334 more