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Horizontal Goodman surgery and almost equivalence of pseudo-Anosov flows

Chi Cheuk Tsang

TL;DR

The paper develops a horizontal generalization of Goodman surgery for (pseudo-)Anosov flows by cutting along transverse annuli and reglueing with a Dehn twist of coefficient 1/n, producing a pseudo-Anosov flow whose orbit structure is controlled away from singular orbits. It proves that for transitive flows, this operation yields an almost equivalence: the surgered flow is orbit-equivalent to the original after drilling finitely many closed orbits, with a structural-stability result underpinning the invariance under choices. A key innovation is the steadiness condition on the foliations used to define the surgery annulus, which enables a cone-field argument to ensure pseudo-Anosovity and a robust analysis near singular orbits, plus a development to piecewise-smooth curves and their approximation by smooth curves. The work connects to broader themes including veering triangulations, Goodman-Fried surgeries, and bicontact structures, and provides concrete corollaries such as mapping-torus twists and scalloped torus constructions, contributing to Fried–Ghys type almost-equivalence questions and expanding the toolkit for constructing and comparing pseudo-Anosov flows.

Abstract

We provide an exposition of a `horizontal' generalization of Goodman's surgery operation on (pseudo-)Anosov flows. This operation is performed by cutting along a specific kind of annulus that is transverse to the flow and regluing with a Dehn twist of the appropriate sign. We then show that performing horizontal Goodman surgery on a transitive pseudo-Anosov flow yields an almost equivalent flow, i.e. the original flow and the surgered flow are orbit equivalent after drilling out a finite collection of closed orbits. We obtain some almost equivalence results by applying this theorem on examples of the surgery operation. Along the way, we also show a structural stability result for pseudo-Anosov flows.

Horizontal Goodman surgery and almost equivalence of pseudo-Anosov flows

TL;DR

The paper develops a horizontal generalization of Goodman surgery for (pseudo-)Anosov flows by cutting along transverse annuli and reglueing with a Dehn twist of coefficient 1/n, producing a pseudo-Anosov flow whose orbit structure is controlled away from singular orbits. It proves that for transitive flows, this operation yields an almost equivalence: the surgered flow is orbit-equivalent to the original after drilling finitely many closed orbits, with a structural-stability result underpinning the invariance under choices. A key innovation is the steadiness condition on the foliations used to define the surgery annulus, which enables a cone-field argument to ensure pseudo-Anosovity and a robust analysis near singular orbits, plus a development to piecewise-smooth curves and their approximation by smooth curves. The work connects to broader themes including veering triangulations, Goodman-Fried surgeries, and bicontact structures, and provides concrete corollaries such as mapping-torus twists and scalloped torus constructions, contributing to Fried–Ghys type almost-equivalence questions and expanding the toolkit for constructing and comparing pseudo-Anosov flows.

Abstract

We provide an exposition of a `horizontal' generalization of Goodman's surgery operation on (pseudo-)Anosov flows. This operation is performed by cutting along a specific kind of annulus that is transverse to the flow and regluing with a Dehn twist of the appropriate sign. We then show that performing horizontal Goodman surgery on a transitive pseudo-Anosov flow yields an almost equivalent flow, i.e. the original flow and the surgered flow are orbit equivalent after drilling out a finite collection of closed orbits. We obtain some almost equivalence results by applying this theorem on examples of the surgery operation. Along the way, we also show a structural stability result for pseudo-Anosov flows.
Paper Structure (32 sections, 37 theorems, 39 equations, 35 figures)

This paper contains 32 sections, 37 theorems, 39 equations, 35 figures.

Table of Contents

  1. Introduction
  2. Pseudo-Anosov flows
  3. Sketch of proof of \ref{['constr:introhorsur']}
  4. A gap in Bar98
  5. Sketch of proof of \ref{['thm:horsuralmostequiv']}
  6. Preliminaries
  7. Pseudo-Anosov flows
  8. Positive/negative curves
  9. Horizontal surgery curves
  10. Definition and properties
  11. Examples
  12. Horizontal Goodman surgery
  13. Surgery operation
  14. Pseudo-Anosovity
  15. Invariance
  16. ...and 17 more sections

Key Result

Theorem 1.3

Let $\phi^t$ be a transitive pseudo-Anosov flow on a closed oriented $3$-manifold $M$. Let $c$ be a positive/negative horizontal surgery curve for $\phi^t$. Then for every positive/negative integer $n$, respectively, the flow $\phi^t_{\frac{1}{n}}(c)$ is almost equivalent to $\phi^t$.

Figures (35)

  • Figure 1: Left: The dynamics of $\phi_{3,0,\lambda}$. Right: A local picture of $\Phi_{3,0,\lambda}$
  • Figure 2: The image of a positive curve (in red) and a negative curve (blue) in the orbit space.
  • Figure 3: Left/right: Local form of a positive/negative surgery curve, respectively, when projected to the orbit space.
  • Figure 4: An annulus $A$ in a positive quadrant of $N$ with one boundary component lying along $\gamma$. In the perspective of the figure, we are looking at $A$ from its back side.
  • Figure 5: We consider the first return maps $f_m$ on a non-separating arc $l$. By the intermediate value theorem, there exists $x \in l$ and $m \in (-\infty,0)$ so that $f_m(x)=x$.
  • ...and 30 more figures

Theorems & Definitions (111)

  • Definition 1.2
  • Theorem 1.3
  • Conjecture 1.4: Fried, Ghys
  • Corollary 1.5
  • Corollary 1.6
  • Corollary 1.7
  • Theorem 1.8
  • Theorem 1.9
  • Definition 2.1
  • Definition 2.3
  • ...and 101 more