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Taut foliations, left-orders, and pseudo-Anosov mapping tori

Jonathan Zung

TL;DR

This work links taut foliations on a broad class of 3-manifolds to nontrivial actions of their fundamental groups on $\mathbb{R}$ by orientation-preserving homeomorphisms, thereby establishing left-orderability in new settings. The authors develop a construction that glues branches of the leaf space of a taut foliation arising from pseudo-Anosov mapping tori, introduces a flat $S^1$-bundle with a flat connection, and then lifts to an $\mathbb{R}$-bundle to realize a nontrivial $\mathbb{R}$-action on the leaf space. Central to the method is a monotone, $\pi_1$-equivariant map from the lifted leaf space to a fiber of $\widehat{\Pi}$, which yields a left-order on $\pi_1$ via standard equivalences. The approach applies to all nontrivial surgeries on the figure-eight knot and yields abundant examples in the Hodgson–Weeks census, significantly expanding known left-orderability results for non-L-space rational homology spheres. The paper also outlines practical computations and poses open questions about extending the framework beyond same-sign slopes and improving the analytic properties of the representations.

Abstract

For a large class of 3-manifolds with taut foliations, we construct an action of $π_1(M)$ on $\mathbb{R}$ by orientation preserving homeomorphisms which captures the transverse geometry of the leaves. This action is complementary to Thurston's universal circle. Applications include the left-orderability of the fundamental groups of every non-trivial surgery on the figure eight knot. Our techniques also apply to at least 2598 manifolds representing 44.7% of the non-L-space rational homology spheres in the Hodgson-Weeks census of small closed hyperbolic 3-manifolds.

Taut foliations, left-orders, and pseudo-Anosov mapping tori

TL;DR

This work links taut foliations on a broad class of 3-manifolds to nontrivial actions of their fundamental groups on by orientation-preserving homeomorphisms, thereby establishing left-orderability in new settings. The authors develop a construction that glues branches of the leaf space of a taut foliation arising from pseudo-Anosov mapping tori, introduces a flat -bundle with a flat connection, and then lifts to an -bundle to realize a nontrivial -action on the leaf space. Central to the method is a monotone, -equivariant map from the lifted leaf space to a fiber of , which yields a left-order on via standard equivalences. The approach applies to all nontrivial surgeries on the figure-eight knot and yields abundant examples in the Hodgson–Weeks census, significantly expanding known left-orderability results for non-L-space rational homology spheres. The paper also outlines practical computations and poses open questions about extending the framework beyond same-sign slopes and improving the analytic properties of the representations.

Abstract

For a large class of 3-manifolds with taut foliations, we construct an action of on by orientation preserving homeomorphisms which captures the transverse geometry of the leaves. This action is complementary to Thurston's universal circle. Applications include the left-orderability of the fundamental groups of every non-trivial surgery on the figure eight knot. Our techniques also apply to at least 2598 manifolds representing 44.7% of the non-L-space rational homology spheres in the Hodgson-Weeks census of small closed hyperbolic 3-manifolds.

Paper Structure

This paper contains 15 sections, 20 theorems, 19 equations, 14 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Notation and conventions
  3. Foliations on surgeries on pseudo-Anosov mapping tori
  4. Structure of branching in $L$
  5. Gluing branches
  6. Hausdorff and non-Hausdorff quotients
  7. Gluing branches in $L$
  8. The flat connection $\widehat \J$
  9. Preliminaries on connections and partial connections
  10. The partial connection $\Jpar$
  11. Blowup time for parallel sections
  12. The full connection $\J$
  13. The fiber of $\widehat\Pi$ and the leaf space
  14. Computations
  15. Remarks and questions

Key Result

Theorem 1.1

Let $\Sigma$ be an orientable closed surface and $\varphi\colon\Sigma\to\Sigma$ a pseudo-Anosov map with orientable invariant foliations. Suppose further that $\varphi$ preserves the orientation of these foliations. Let $M_\varphi$ be the mapping torus of $\varphi$. Let $M_\varphi(\textbf{p};\textbf

Figures (14)

  • Figure 1: This figure shows a train track carrying an invariant lamination for the monodromy of the fibered, 1-cusped 3-manifold m038. The fiber surface has genus 2 with one puncture at the vertices of the octagon. The complement of the train track is a punctured ideal hexagon. The figure was generated using Mark Bell's program flipperbell_flipper_2013.
  • Figure 2: Blowing down $\mathcal{I}$. The shaded region on the left is $\mathcal{G}$, and the shaded region on the right is $\mathcal{G}'$.
  • Figure 3: On the left is the standard foliation of $D^2\times S^1$, where $S^1$ is cut open. In the middle, we alternately comb the edges of the disks to expose their positive and negative sides. Grey shows the positive sides of the leaves, while white shows the negative sides. On the right, we show the limiting configuration which has 4 annular leaves at the boundary which we call walls and infinitely tall saddle-like leaves (homeomorphic to planes) on the interior. The interior saddle-like leaves accumulate on the walls. We can vary the number of legs of the saddle's rider or the gluing of the top and bottom of the picture to get sutures of any desired non-meridional slope.
  • Figure 4: Various tightening moves
  • Figure 7: Two paths that are to be glued together, $[v_0,v_0LRRR\dots)$ and $[v_0,v_0RLRRR\dots)$, are shown in red and blue respectively. Dotted lines are drawn through the equivalence classes $2$, $2'$, and $2"$. These classes map to non-separable points in the quotient. Their canonical representatives are $v_0LL$, $v_0RLL$, and $v_0RRLL$. For any $s\geq 0$, the point $v_0LR^s$ (marked with the label $2-\frac{1}{2^s}$) is a common ancestor of all three of these points in the quotient. Moreover, as $s\to \infty$, $v_0LR^s$ converges to all three of these points, so the space is not Hausdorff.
  • ...and 9 more figures

Theorems & Definitions (54)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Example 2.1
  • proof : Proof of \ref{['cor:fig8']}
  • Example 2.2
  • Example 2.3
  • Lemma 2.5
  • proof
  • Example 3.3
  • ...and 44 more