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Non R-covered Anosov flows in hyperbolic 3-manifolds are quasigeodesic

Sergio R Fenley

Abstract

The main result is that if an Anosov flow in a closed hyperbolic three manifold is not R-covered, then the flow is a quasigeodesic flow. We also prove that if a hyperbolic three manifold supports an Anosov flow, then up to a double cover it supports a quasigeodesic flow. We prove the continuous extension property for the stable and unstable foliations of any Anosov flow in a closed hyperbolic three manifold, and the existence of group invariant Peano curves associated with any such flow.

Non R-covered Anosov flows in hyperbolic 3-manifolds are quasigeodesic

Abstract

The main result is that if an Anosov flow in a closed hyperbolic three manifold is not R-covered, then the flow is a quasigeodesic flow. We also prove that if a hyperbolic three manifold supports an Anosov flow, then up to a double cover it supports a quasigeodesic flow. We prove the continuous extension property for the stable and unstable foliations of any Anosov flow in a closed hyperbolic three manifold, and the existence of group invariant Peano curves associated with any such flow.
Paper Structure (29 sections, 43 theorems, 28 equations, 22 figures)

This paper contains 29 sections, 43 theorems, 28 equations, 22 figures.

Table of Contents

  1. Introduction
  2. Examples of $\hbox{${\mathbb R}$}$-covered and non $\hbox{${\mathbb R}$}$-covered Anosov flows in hyperbolic $3$-manifolds
  3. Some ideas on the proof of the Main Theorem
  4. Brief description of sections
  5. Acknowledgements
  6. Background on topological Anosov flows
  7. Limits of lozenges and canonical bands
  8. Blocks
  9. Good form of the flow $\Phi$
  10. More information about the proof of the Main theorem
  11. Bands
  12. Limits of lozenges
  13. Bands and finite adjacent blocks
  14. Limits of bands of lozenges
  15. Construction of walls
  16. ...and 14 more sections

Key Result

Theorem 1.1

Let $M$ be a hyperbolic $3$-manifold admitting an Anosov flow. Then up to perhaps a double cover, $M$ admits a quasigeodesic pseudo-Anosov flow. In any case $M$ admits a one dimensional foliation by quasigeodesics, with a dense leaf.

Figures (22)

  • Figure 1: a. A perfect fit. The half leaves $F_1, G_1$ of $F, G$ depict the perfect fit between $F, G$. Here $F_1, G_1$ do not intersect, but any $S$ intersecting $L$ between $G$ and $H$ intersects $F_1$;, b. A lozenge. $p, q$ are the corners of the lozenge; c. A chain of lozenges.
  • Figure 2: A chain of adjacent lozenges $\{ A_i, 1 \leq i \leq n \}$ connecting non separated leaves $F, L$ in $\widetilde{\Lambda}^s$. Here $F_0$ is a half leaf of $F$ contained in the boundary of $A_1$ and similarly $L_0$ is a half leaf of $L$ contained in the boundary of $A_6$.
  • Figure 3: A finite block. Assuming that $S$ is a stable leaf, then this is a stable adjacent block. It has 5 elements, $C_1, ...., C_5$. The element $C_1$ is a lozenge, and $C_2$ through $C_5$ are $(1,3)$ ideal quadrilaterals.
  • Figure 4: A more involved finite block. We assume that $S$ is a stable leaf and $U$ is an unstable leaf. The block is made up of 10 elements $C_1, ..., C_7$ and $D_1, D_2, D_3$. The $C's$ form a stable adjacent block with one lozenge and 6 other elements which are $(1,3)$ ideal quadrilaterals. In this example in this stable adjacent block the lozenge is neither the first, nor the last element in this block. The $D's$ form an unstable adjacent block, with one lozenge $D_3$ and two $(1,3)$ ideal quadrilaterals, $D_1, D_2$.
  • Figure 5: A band $B$. The figure depicts the foliation $\hbox{$\widetilde{\Lambda}^s$} \cap B$, with its geometric behavior: the leaves in the interior are asymptotic to the boundary leaves in the negative flow direction. The arrows in $\ell_0$ and $\ell_1$ indicate the flow direction in these flow lines.
  • ...and 17 more figures

Theorems & Definitions (116)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Remark 1.4
  • Remark 1.5
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • ...and 106 more