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Depth-one foliations, pseudo-Anosov flows and universal circles

Junzhi Huang

Abstract

Given a taut depth-one foliation $\mathcal{F}$ in a closed atoroidal 3-manifold $M$ transverse to a pseudo-Anosov flow $φ$ without perfect fits, we show that the universal circle coming from leftmost sections $\mathfrak{S}_\mathrm{left}$ associated to $\mathcal{F}$, constructed by Thurston and Calegari-Dunfield, is isomorphic to the ideal boundary of the flow space associated to $φ$ with natural structure maps. As a corollary, we use a theorem of Barthelmé-Frankel-Mann to show that there is at most one pseudo-Anosov flow without perfect fits transverse to $\mathcal{F}$ up to orbit equivalence.

Depth-one foliations, pseudo-Anosov flows and universal circles

Abstract

Given a taut depth-one foliation in a closed atoroidal 3-manifold transverse to a pseudo-Anosov flow without perfect fits, we show that the universal circle coming from leftmost sections associated to , constructed by Thurston and Calegari-Dunfield, is isomorphic to the ideal boundary of the flow space associated to with natural structure maps. As a corollary, we use a theorem of Barthelmé-Frankel-Mann to show that there is at most one pseudo-Anosov flow without perfect fits transverse to up to orbit equivalence.

Paper Structure

This paper contains 10 sections, 33 theorems, 21 equations, 15 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Pseudo-Anosov flows
  4. End-periodic automorphisms
  5. Depth-one foliation
  6. Laminations on $S^1$
  7. Infinity of shadows
  8. Markers and universal circles
  9. Building the homeomorphism
  10. Invariant laminations

Key Result

Theorem 1.1

Let $M$ be a closed atoroidal 3-manifold with a pseudo-Anosov flow $\phi$ without perfect fits, and let $\mathcal{F}$ be a taut depth-one foliation in $M$ transverse to $\phi$. Then the circle $\partial\mathcal{O}$, together with the structure maps $\{I_\lambda\}_{\lambda\in\widetilde{\mathcal{F}}}$

Figures (15)

  • Figure 1:
  • Figure 2: Type-1 leaves (black) in a product region limit to type-0 leaves (blue) in the positive boundary and stay transverse to $\widetilde{\phi}$ (red), creating non-Hausdorff-ness in $\Lambda$
  • Figure 3: Left: a local picture near a product region $\widetilde{\Omega}$ in $\Lambda$. Right: the corresponding parts in $\Lambda^*$. The red vertices represent type-0 leaves in both pictures, and the arrows indicate the direction of $\widetilde{\phi}$.
  • Figure 4:
  • Figure 5: The vertical arrowed line represents the periodic orbit $\sigma_e$, the red plane is $\widetilde{\mathcal{F}}^s(\sigma_e)$, the blue plane is $\widetilde{\mathcal{F}}^u(\sigma_e)$, and the green plane is $\lambda$.
  • ...and 10 more figures

Theorems & Definitions (65)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • proof
  • Theorem 1.4
  • Lemma 2.2
  • Proposition 3.1: Fenley1999823
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • ...and 55 more