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Completing Prelaminations

Thomas Barthelmé, Christian Bonatti, Kathryn Mann

Abstract

Motivated by problems in the study of Anosov and pseudo-Anosov flows on 3-manifolds, we characterize when a pair $(L^+, L^-)$ of subsets of transverse laminations of the circle can be completed to a pair of transverse foliations of the plane or, separately, realized as the endpoints of such a bifoliation of the plane. (We allow also singular bifoliations with simple prongs, such as arise in pseudo-Anosov flows). This program is carried out at a level of generality applicable to bifoliations coming from pseudo-Anosov flows with and without perfect fits, as well as many other examples, and is natural with respect to group actions preserving these structures.

Completing Prelaminations

Abstract

Motivated by problems in the study of Anosov and pseudo-Anosov flows on 3-manifolds, we characterize when a pair of subsets of transverse laminations of the circle can be completed to a pair of transverse foliations of the plane or, separately, realized as the endpoints of such a bifoliation of the plane. (We allow also singular bifoliations with simple prongs, such as arise in pseudo-Anosov flows). This program is carried out at a level of generality applicable to bifoliations coming from pseudo-Anosov flows with and without perfect fits, as well as many other examples, and is natural with respect to group actions preserving these structures.

Paper Structure

This paper contains 22 sections, 76 theorems, 7 equations, 5 figures.

Table of Contents

  1. Introduction
  2. An introduction to bifoliations and (pre)-laminations
  3. Results
  4. Outline
  5. Proof of Theorem \ref{['thm:realization']}: Constructing the plane
  6. The crossing space
  7. The polygon topology
  8. $\mathcal{P}$ is a bifoliated plane
  9. Circle at infinity
  10. Uniqueness
  11. Converse
  12. A toolkit for completing prelaminations to bifoliations
  13. Linkage graphs and their basic properties
  14. Adding leaves to prelaminations
  15. Crossing geodesics
  16. ...and 7 more sections

Key Result

Theorem 1.2

Let $L^+, L^-$ be regular subsets of laminations of $S^1$, with no shared leaves. This pair can be completed to a non-singular bifoliation of the plane if and only if the following two conditions hold: Moreover, this completion is unique up to foliation-preserving homeomorphisms restricting to the identity on $S^1$, and any group acting on $S^1$ preserving $L^+, L^-$ extends to act (uniquely) b

Figures (5)

  • Figure 1: A one-root region and a coupled pair of ideal polygons
  • Figure 2: A complementary region (in blue) and its linkage graph. Note the ideal side divided into three ideal segments giving three vertices. See figures \ref{['fig:condition_4']} or \ref{['fig:crossing']} for examples with a cycle.
  • Figure 3: An example of a high valence leaf (blue, diagonal) satisfying \ref{['item_high_val_two_polygons']} but none of the other conditions, with the linkage graphs for the two complementary regions of the blue foliation shown.
  • Figure 4: Crossing edges (dotted lines) associated to a valence 1 cut-edge, and to a cut-pair. The original linkage graph (before adding crossing edges), and resulting three graphs (after) are shown.
  • Figure 5: Two (partially defined) prelaminations that are homeomorphic on $S^1$ but with no extension to $\mathbb D^2$.

Theorems & Definitions (198)

  • Definition 1.1
  • Theorem 1.2: Special case of main theorem
  • Theorem 1: Main theorem -- completions
  • Theorem 2: Extending group actions
  • Definition 1.3
  • Definition 1.4
  • Definition 1.5
  • Definition 1.6
  • Definition 1.7
  • Definition 1.8
  • ...and 188 more