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Orbital equivalence classes of finite coverings of geodesic flows

Thierry Barbot, Sérgio Fenley

Abstract

Let $M$ be a closed 3-manifold admitting a finite cover of index n along the fibers over the unit tangent bundle of a closed surface. We prove that if n is odd, there is only one Anosov flow on M up to orbital equivalence, and if n is even, there are two orbital equivalence classes of Anosov flows on M.

Orbital equivalence classes of finite coverings of geodesic flows

Abstract

Let be a closed 3-manifold admitting a finite cover of index n along the fibers over the unit tangent bundle of a closed surface. We prove that if n is odd, there is only one Anosov flow on M up to orbital equivalence, and if n is even, there are two orbital equivalence classes of Anosov flows on M.
Paper Structure (14 sections, 17 theorems, 82 equations, 6 figures)

This paper contains 14 sections, 17 theorems, 82 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Anosov flows $-$ definitions
  4. Orbit space and leaf spaces of Anosov flows
  5. Recovering the Anosov flow from the orbit space
  6. Classification of finite coverings of geodesic flows up to isotopy and orbital equivalence
  7. Geodesic flows
  8. Geodesic flows as algebraic flows.
  9. Geodesic flow on the space of triples.
  10. Geodesic flows on the projectived tangent bundle over the band.
  11. Anosov flows on the unit tangent bundle over a closed surface
  12. Isotopy classes of Anosov flows on circle bundles over closed surfaces
  13. Anosov flows on circle bundles over closed surfaces up to orbital equivalence
  14. Counting the number of orbits of the action of the modular group on the set of finite index subgroups

Key Result

Theorem 2.4

(Ba2) Let $(M, \Phi^t)$ be an $\mathbb{R}$-covered Anosov flow. Assume that $(M, \Phi^t)$ is not orbitally equivalent to the suspension of an Anosov diffeomorphism of the torus. Then, the map $\Upsilon: \hbox{$\mathcal{O}$} \to Q^s \times Q^u$ sending an orbit $\tilde{\theta}$ to the pair $(\hbox{${ where $\alpha$ and $\beta$ are two homeomorphisms from $Q^s$ into $Q^u$ satisfying, for every $\gam

Figures (6)

  • Figure 1: A collection of oriented loops $(\bar{a}_1, \bar{b}_1, \ldots , \bar{a}_g, \bar{b}_g)$ forming a symplectic basis in homology, here for $g=4$. The surface is considered has the boundary of an handlebody in $\mathbb{R}^3$ and the orientation of the surface is given by the normal vector pointing out the handlebody.
  • Figure 2: According to Lickorish's Theorem, Dehn twists along these simple closed curves generate the mapping class group.
  • Figure 3: Definition of $\bar{a}'_2$, $\bar{a}'_3$, $\bar{c}'_2$.
  • Figure 4: $\Delta$, $\Delta_i$ and $\Delta_{i+1}$ are preserved by respectively $\bar{c}^{-1}$, $\bar{a}_i$ and $(\bar{a}"_{i+1})^{-1}$. The attracting fixed point of $(\bar{a}"_{i+1})^{-1}$ is $x_1$, and the attracting fixed point of $\bar{a}_i$ is $x_2$.
  • Figure 5: Liftings in $\widetilde{\mathbb{R}P}^1$. We have depicted two attracting fixed points $\tilde{x}_1$ and $h(\tilde{x}_1)$ of $({a}"_{i+1})^{-1}$, and two attracting fixed points $\tilde{x}_2$ and $h^{-1}(\tilde{x}_2)$ of ${a}_i$.
  • ...and 1 more figures

Theorems & Definitions (46)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Theorem 2.4
  • Remark 2.5
  • Remark 2.6
  • Remark 2.7
  • Corollary 2.8
  • proof
  • Theorem 2.9
  • ...and 36 more