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Left orderability and taut foliations with one-sided branching

Bojun Zhao

Abstract

For a closed orientable irreducible $3$-manifold $M$ that admits a co-orientable taut foliation with one-sided branching, we show that $π_1(M)$ is left orderable.

Left orderability and taut foliations with one-sided branching

Abstract

For a closed orientable irreducible -manifold that admits a co-orientable taut foliation with one-sided branching, we show that is left orderable.
Paper Structure (11 sections, 14 theorems, 25 equations, 2 figures)

This paper contains 11 sections, 14 theorems, 25 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Acknowledgements
  3. Preliminary
  4. The leaf space of a taut foliation
  5. $1$-manifolds with one-sided branching
  6. The three types of foliations
  7. The blowing-up operations
  8. The group $\mathcal{G}_\infty$
  9. The proof of the main theorem
  10. The group action on a $1$-manifold with one-sided branching
  11. The proof of Theorem \ref{['foliation']}

Key Result

Theorem 1.1

Let $M$ be a connected, closed, orientable, irreducible $3$-manifold that admits a co-orientable taut foliation with one-sided branching. Then $\pi_1(M)$ is left orderable.

Figures (2)

  • Figure 1: Picture (a) describes a collection of points in $L(\mathcal{F})$ which are pairwise non-separated. Picture (b) describes the ends of $L(\mathcal{F})$: every positive end of $L(\mathcal{F})$ can be represented by some positively oriented rays in $L(\mathcal{F})$, and every negative end of $L(\mathcal{F})$ can be represented by some negatively oriented rays in $L(\mathcal{F})$.
  • Figure 2: For $g \in G$ with $ge(\mathbb{R}) \ne e(\mathbb{R})$, there are $t,r \in \mathbb{R}$ such that the image of $(t,+\infty)$ under $e$ is same as the image of $(r,+\infty)$ under $ge$, but $e(t) \ne ge(r)$

Theorems & Definitions (32)

  • Theorem 1.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Lemma 2.5
  • proof
  • Definition 2.6
  • Lemma 2.7
  • proof
  • Corollary 2.8
  • ...and 22 more