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From veering triangulations to link spaces and back again

Steven Frankel, Saul Schleimer, Henry Segerman

TL;DR

This work constructs a canonical link-space framework and veering circle from a veering triangulation, establishing a concrete dictionary between veering combinatorics and flow dynamics on 3-manifolds. By collapsing invariant laminations in the veering circle, the authors obtain the link space, prove it is a loom space, and show the functorial relationship between veering triangulations and loom spaces; in the reverse direction, dynamics on the link space canonically recover the veering triangulation. They also outline a Cannon–Thurston-type program, linking veering circles to hyperbolic boundaries via convergence-group actions, and position these results within a broader program unifying combinatorial 3-manifold topology with pseudo-Anosov dynamics. The paper solidifies the first half of a bijection between veering data and flow-box decompositions without perfect fits, setting up a robust framework for comparing dynamics and triangulations and enabling subsequent algorithmic and geometric applications. The results have significant implications for understanding surgery parents of pseudo-Anosov flows and for constructing canonical invariants in 3-manifold topology.

Abstract

This paper is the third in a sequence establishing a dictionary between the combinatorics of veering triangulations equipped with appropriate filling slopes, and the dynamics of pseudo-Anosov flows (without perfect fits) on closed three-manifolds. Our motivation comes from the work of Agol and Guéritaud. Agol introduced veering triangulations of mapping tori as a tool for understanding the surgery parents of pseudo-Anosov mapping tori. Guéritaud gave a new construction of veering triangulations of mapping tori using the orbit spaces of their suspension flows. Generalising this, Agol and Guéritaud announced a method that, given a closed manifold with a pseudo-Anosov flow (without perfect fits), produces a veering triangulation equipped with filling slopes. In this paper we build, from a veering triangulation, a canonical circular order on the cusps of the universal cover. Using this we build the veering circle and the link space. These are the first entries in the promised dictionary. The link space and the circle are, respectively, analogous to the orbit space of a flow and to Fenley's boundary at infinity of the orbit space. In the other direction, and using our previous work, we prove that the veering triangulation is recovered (up to canonical isomorphism) from the dynamics of the fundamental group acting on the link space. This is the first step in proving that our dictionary gives a bijection between the two theories.

From veering triangulations to link spaces and back again

TL;DR

This work constructs a canonical link-space framework and veering circle from a veering triangulation, establishing a concrete dictionary between veering combinatorics and flow dynamics on 3-manifolds. By collapsing invariant laminations in the veering circle, the authors obtain the link space, prove it is a loom space, and show the functorial relationship between veering triangulations and loom spaces; in the reverse direction, dynamics on the link space canonically recover the veering triangulation. They also outline a Cannon–Thurston-type program, linking veering circles to hyperbolic boundaries via convergence-group actions, and position these results within a broader program unifying combinatorial 3-manifold topology with pseudo-Anosov dynamics. The paper solidifies the first half of a bijection between veering data and flow-box decompositions without perfect fits, setting up a robust framework for comparing dynamics and triangulations and enabling subsequent algorithmic and geometric applications. The results have significant implications for understanding surgery parents of pseudo-Anosov flows and for constructing canonical invariants in 3-manifold topology.

Abstract

This paper is the third in a sequence establishing a dictionary between the combinatorics of veering triangulations equipped with appropriate filling slopes, and the dynamics of pseudo-Anosov flows (without perfect fits) on closed three-manifolds. Our motivation comes from the work of Agol and Guéritaud. Agol introduced veering triangulations of mapping tori as a tool for understanding the surgery parents of pseudo-Anosov mapping tori. Guéritaud gave a new construction of veering triangulations of mapping tori using the orbit spaces of their suspension flows. Generalising this, Agol and Guéritaud announced a method that, given a closed manifold with a pseudo-Anosov flow (without perfect fits), produces a veering triangulation equipped with filling slopes. In this paper we build, from a veering triangulation, a canonical circular order on the cusps of the universal cover. Using this we build the veering circle and the link space. These are the first entries in the promised dictionary. The link space and the circle are, respectively, analogous to the orbit space of a flow and to Fenley's boundary at infinity of the orbit space. In the other direction, and using our previous work, we prove that the veering triangulation is recovered (up to canonical isomorphism) from the dynamics of the fundamental group acting on the link space. This is the first step in proving that our dictionary gives a bijection between the two theories.

Paper Structure

This paper contains 75 sections, 91 theorems, 59 equations, 60 figures.

Table of Contents

  1. Introduction
  2. The dictionary
  3. Link spaces
  4. Dynamic pairs
  5. Flow-box decompositions
  6. The correspondence
  7. Cannon--Thurston maps
  8. Veering triangulations
  9. Outline
  10. Circular orders and taut ideal triangulations
  11. Circular orders
  12. Taut ideal triangulations
  13. Horizontal branched surface
  14. Layered triangulations
  15. Compatibility
  16. ...and 60 more sections

Key Result

Theorem 2.12

Suppose that $M$ is a three-manifold equipped with a taut ideal triangulation $\mathcal{T}$. Suppose that $t$ is a model tetrahedron of ${\widetilde{\mathcal{T}}}$. Then $t$ embeds in ${\widetilde{M}}$; furthermore, the model vertices of $t$ lie in four distinct cusp of $\Delta_\mathcal{T}$.

Figures (60)

  • Figure 1.1: The veering structure on the canonical triangulation for the figure-eight knot complement.
  • Figure 1.1: The bigon track $\tau(c)$ in the vertex link $L(c) \subset N(c)$, for the SnapPy manifold s227 as determined by the veering triangulation gLLAQbecdfffhhnkqnc_120012 from the veering census GSS19. Note that we only give a fundamental domain for the action of $\operatorname{Stab}(c)$. Here the action of $\operatorname{Stab}(c)$ is generated by two translations parallel to the $x$-- and $y$--axes, respectively. The colours and orientations are discussed in the main text.
  • Figure 1.2: The main objects and constructions in this paper.
  • Figure 1.2: The strip $P$ turns right and then left immediately before and after, respectively, the edge $e"_n$.
  • Figure 1.3:
  • ...and 55 more figures

Theorems & Definitions (309)

  • Remark 1.2
  • Remark 1.3
  • Remark 1.7
  • Definition 2.2
  • Example 2.3
  • Definition 2.4
  • Definition 2.5
  • Example 2.6
  • Example 2.7
  • Remark 2.8
  • ...and 299 more