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The triangulation complexity of fibred 3-manifolds

Marc Lackenby, Jessica S. Purcell

Abstract

The triangulation complexity of a closed orientable 3-manifold is the minimal number of tetrahedra in any triangulation of the manifold. The main theorem of the paper gives upper and lower bounds on the triangulation complexity of any closed orientable hyperbolic 3-manifold that fibres over the circle. We show that the triangulation complexity of the manifold is equal to the translation length of the monodromy action on the mapping class group of the fibre, up to a bounded factor, where the bound depends only on the genus of the fibre.

The triangulation complexity of fibred 3-manifolds

Abstract

The triangulation complexity of a closed orientable 3-manifold is the minimal number of tetrahedra in any triangulation of the manifold. The main theorem of the paper gives upper and lower bounds on the triangulation complexity of any closed orientable hyperbolic 3-manifold that fibres over the circle. We show that the triangulation complexity of the manifold is equal to the translation length of the monodromy action on the mapping class group of the fibre, up to a bounded factor, where the bound depends only on the genus of the fibre.

Paper Structure

This paper contains 36 sections, 87 theorems, 25 equations, 34 figures.

Table of Contents

  1. Introduction
  2. Translation length and analogous results
  3. Main results
  4. Applications
  5. Further work
  6. Outline of proof
  7. Acknowledgements
  8. The spine graph and triangulation graphs
  9. The spine graph
  10. Edge contractions and expansions on the same surface
  11. Stable translation length in the mapping class group
  12. Train tracks
  13. Splits and slides
  14. pseudo-Anosov homeomorphisms
  15. Eventually periodic train track sequences
  16. ...and 21 more sections

Key Result

Theorem 1.1

Let $S$ be a compact orientable surface. Then the following quantities are all within bounded ratios of each other, where the bounds only depend on the Euler characteristic of $S$, for a pseudo-Anosov homeomorphism $\phi$ of $S$:

Figures (34)

  • Figure 1: An edge contraction/expansion
  • Figure 2: 2-2 and 1-3 Pachner moves
  • Figure 3: Top: The three types of split; Bottom: A slide
  • Figure 4: Normal and almost normal pieces, left to right: triangle, square, octagon, tubed piece.
  • Figure 5: A handle structure arising from a triangulation. Shown also is a shaded surface that respects the handle structure.
  • ...and 29 more figures

Theorems & Definitions (224)

  • Theorem 1.1: Brock
  • Theorem 1.2: Futer--Schleimer
  • Theorem 1.3
  • Theorem 1.4
  • Corollary 1.5
  • Theorem 1.6
  • Conjecture 1.7
  • Definition 2.1
  • Lemma 2.2: Bound on vertices and edges, spine
  • proof
  • ...and 214 more