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Left-orderability in Dehn fillings of pseudo-Anosov mapping tori

Bojun Zhao

Abstract

For pseudo-Anosov mapping tori with co-orientable invariant foliations and monodromies reversing their co-orientations, a family of taut foliations was constructed in previous work on Dehn fillings with all rational slopes outside a neighborhood of the degeneracy slope. In this paper, we prove that all such Dehn fillings have left-orderable fundamental groups. We present two approaches, both based on an analysis of the branching behavior from such taut foliations. The first approach produces an $\mathbb{R}$-covered foliation arising from this family for each filling slope, and the second approach shows that, depending on the choice of a suitable system of arcs on $Σ$, one obtains a foliation that either has one-sided branching or is $\mathbb{R}$-covered. Consequently, the second approach associates to each Dehn filling a family of representations of its fundamental group into $\mathcal{G}_\infty$, the group of germs at infinity, whereas the first approach yields an explicit left-invariant order. As an application, combining our results with earlier work in the literature, we verify the L-space conjecture for all surgeries on the $(-2,3,2k+1)$-pretzel knot ($k \geqslant 3$) in $S^3$.

Left-orderability in Dehn fillings of pseudo-Anosov mapping tori

Abstract

For pseudo-Anosov mapping tori with co-orientable invariant foliations and monodromies reversing their co-orientations, a family of taut foliations was constructed in previous work on Dehn fillings with all rational slopes outside a neighborhood of the degeneracy slope. In this paper, we prove that all such Dehn fillings have left-orderable fundamental groups. We present two approaches, both based on an analysis of the branching behavior from such taut foliations. The first approach produces an -covered foliation arising from this family for each filling slope, and the second approach shows that, depending on the choice of a suitable system of arcs on , one obtains a foliation that either has one-sided branching or is -covered. Consequently, the second approach associates to each Dehn filling a family of representations of its fundamental group into , the group of germs at infinity, whereas the first approach yields an explicit left-invariant order. As an application, combining our results with earlier work in the literature, we verify the L-space conjecture for all surgeries on the -pretzel knot () in .

Paper Structure

This paper contains 19 sections, 23 theorems, 56 equations, 10 figures.

Table of Contents

  1. Introduction
  2. Left-orderability and Dehn fillings on cusped manifolds
  3. Applications in specific knots
  4. Foliations with at most one-sided branching
  5. Organization
  6. Acknowledgement
  7. Preliminaries
  8. Conventions
  9. Branching behaviors of taut foliations
  10. Pseudo-Anosov flows
  11. The branched surface
  12. The foliation $\mathcal{F}_\alpha$ transverse to the suspension flow
  13. Producing $\mathbb{R}$-covered foliations
  14. A Refined admissible system of arcs
  15. The intersection behavior of $\mathcal{F}_\alpha$ and the weak stable foliation
  16. ...and 4 more sections

Key Result

Theorem 1.3

Suppose that $\mathcal{F}^{s}$ is co-orientable and $\varphi$ reverses its co-orientation. For each $1 \leqslant i \leqslant r$, let $J_i$ be the open interval in $\mathbb{R} \cup \{\infty\} \cong \mathbb{R} P^{1}$ between $\frac{p_i}{q_i + c_i}$ and $\frac{p_i}{q_i - c_i}$ which does not contain $\

Figures (10)

  • Figure 1: For a Floer simple knot manifold, there exists a finite set $P$ determined by the Turaev torsion such that a rational slope yields an L-space Dehn filling if and only if it lies in the closure of a component of $(\mathbb{R} \cup \{\infty\}) - P$RasmussenR17. In (a), the set $P$ is shown as blue dots. The dashed segment represents the closed interval of L-space slopes, and every rational slope in its complement (the solid segment) yields a non-L-space filling. In (b), we illustrate the slopes appearing in Theorem \ref{['foliation']}. The degeneracy slope $\frac{p_i}{q_i}$ is shown as the red dot, the two bounds $\frac{p_i}{q_i \pm c_i}$ as blue dots, the interval $J_i$ as the solid segment, and the remaining neighborhood of the degeneracy slope as the dashed segment.
  • Figure 2: Local models of a standard spine in a $3$-manifold $M$ of various types. In (d) and (e), the standard spine has boundary on $\partial M$, and the shaded regions lie in $\partial M$.
  • Figure 3: Local models of a branched surface in a $3$-manifold $M$ of various types. In (d) and (e), the branched surface has boundary on $\partial M$, which is shown shaded.
  • Figure 4: (a) A local model of a branched surface $B$. (b) The corresponding local model of a fibered neighborhood $N(B)$ of $B$.
  • Figure 5: (a) A local picture of the arc $\alpha^+_i$ obtained from $\alpha_i$ by isotopy, and the band $B_i$ represented by the shaded region. The red segments represent leaves of $\mathcal{F}^{s}$. (b) A local model of $N(B_\alpha)$, whose interval fibers are contained in orbits of $\psi$.
  • ...and 5 more figures

Theorems & Definitions (55)

  • Conjecture 1: L-space conjecture, BGW13Juhasz15
  • Theorem 1.3: Z26
  • Theorem 1.4
  • Corollary 1.5
  • Corollary 1.6
  • Proposition 1.7
  • Proposition 1.8
  • Proposition 1.9
  • Example 1.10
  • Example 1.11
  • ...and 45 more