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Zippers

Danny Calegari, Ino Loukidou

Abstract

If $M$ is a hyperbolic 3-manifold fibering over the circle, the fundamental group of $M$ acts faithfully by homeomorphisms on a circle (the circle at infinity of the universal cover of the fiber), preserving a pair of invariant (stable and unstable) laminations. Many different kinds of dynamical structures (e.g. taut foliations, quasigeodesic or pseudo-Anosov flows) are known to give rise to universal circles -- a circle with a faithful $π_1(M)$ action preserving a pair of invariant laminations -- and these universal circles play a key role in relating the dynamical structure to the geometry of $M$. In this paper we introduce the idea of zippers, which give a new and direct way to construct universal circles, streamlining the known constructions in many cases, and giving a host of new constructions in others. In particular, zippers (and their associated universal circles) may be constructed directly from uniform quasimorphisms or from uniform left orders.

Zippers

Abstract

If is a hyperbolic 3-manifold fibering over the circle, the fundamental group of acts faithfully by homeomorphisms on a circle (the circle at infinity of the universal cover of the fiber), preserving a pair of invariant (stable and unstable) laminations. Many different kinds of dynamical structures (e.g. taut foliations, quasigeodesic or pseudo-Anosov flows) are known to give rise to universal circles -- a circle with a faithful action preserving a pair of invariant laminations -- and these universal circles play a key role in relating the dynamical structure to the geometry of . In this paper we introduce the idea of zippers, which give a new and direct way to construct universal circles, streamlining the known constructions in many cases, and giving a host of new constructions in others. In particular, zippers (and their associated universal circles) may be constructed directly from uniform quasimorphisms or from uniform left orders.

Paper Structure

This paper contains 34 sections, 24 theorems, 6 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Zippers and Uniforms
  3. The $L$-space conjecture
  4. Statement of Results
  5. Zippers
  6. Definition
  7. Basic properties
  8. Landing rays
  9. Circular orders
  10. Ends and ideal gaps
  11. End circles
  12. Bridges and the universal circle
  13. Invariant laminations
  14. Main Theorem
  15. P-zippers
  16. ...and 19 more sections

Key Result

Lemma 2.3

Each of $Z^\pm$ is dense in $S^2_\infty$.

Figures (6)

  • Figure 1: A zipper associated to the $(0,2)$ orbifold filling on the Figure 8 knot complement.
  • Figure 2: There are unique disjoint paths (in blue) from $K$ (in black) to the three ends $e_1,e_2,e_3$; these intersect $\partial D$ (in green) for the first time in three distinct points (in red); the cyclic order of these three points in $\partial D$ gives the circular order on $e_1,e_2,e_3$.
  • Figure 3: $J$ (in red) is a path from $p_0$ (in green) to $X$ (in orange) representing the equivalence class of an ideal gap.
  • Figure 4: $gh^{-1}$ has an axis.
  • Figure 5: The staircases $a$ and $b$ may be interpolated by successively filling in blocks.
  • ...and 1 more figures

Theorems & Definitions (93)

  • Definition 2.1
  • Remark 2.2
  • Lemma 2.3: Dense
  • proof
  • Lemma 2.4: Unique path
  • proof
  • Definition 2.5: Convex hull
  • Definition 2.6: Path topology
  • Definition 2.7: Hard end
  • Definition 2.8: End space
  • ...and 83 more