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Polynomially many surfaces of fixed Euler characteristic in a hyperbolic 3-manifold

Marc Lackenby, Anastasiia Tsvietkova

Abstract

We give an upper bound for the number of compact essential orientable non-isotopic surfaces, with Euler characteristic at least some constant $χ$, properly embedded in a finite-volume hyperbolic 3-manifold $M$, closed or cusped. This bound is a polynomial function of the volume of $M$, with degree that depends linearly on $|χ|$.

Polynomially many surfaces of fixed Euler characteristic in a hyperbolic 3-manifold

Abstract

We give an upper bound for the number of compact essential orientable non-isotopic surfaces, with Euler characteristic at least some constant , properly embedded in a finite-volume hyperbolic 3-manifold , closed or cusped. This bound is a polynomial function of the volume of , with degree that depends linearly on .
Paper Structure (19 sections, 47 theorems, 37 equations, 9 figures)

This paper contains 19 sections, 47 theorems, 37 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Related work
  3. Acknowledgements
  4. Overview of the proof
  5. Triangulating a fat part of the 3-manifold
  6. Thick-thin decomposition.
  7. A fat part of $M$
  8. Triangulating a fat part of $M$ with a 'thick' triangulation.
  9. Stable minimal surfaces
  10. Background.
  11. A survey of relevant results
  12. Consequences of minimality
  13. Consequences of stability
  14. Triangulations transverse to a surface
  15. Barycentric subdivision
  16. ...and 4 more sections

Key Result

Theorem 1.1

There are constants $c_1$ and $c_2$ with the following property. Let $M$ be an orientable hyperbolic 3-manifold of finite volume $\mathrm{vol}(M)$, closed or with cusps. The number of properly embedded orientable essential surfaces in $M$ with Euler characteristic at least $\chi$, up to isotopy, is

Figures (9)

  • Figure 1: A normal disc in a tetrahedron is a triangle or square, as shown.
  • Figure 2: Shown is the intersection between $B(0, \eta)$ and the plane $P$ through $0$, $p_1$ and $q_1$.
  • Figure 3: The annuli $A_1$ and $A_2$ in $\partial B(0,\eta)$. If the angle between $n_1$ and $n_2$ were greater than $\delta"$, then $A_1 \cap A_2$ consists of two discs.
  • Figure 4: A simple closed curve $C$ of $R \cap F$ corresponds to a simple closed curve $\tilde{C}$ of $\tilde{R} \cap \tilde{F}$. Some unit tangent vector of $\tilde{C}$ is almost parallel to a unit tangent vector at some point $\tilde{y}$ on some edge of $\tilde{R}$.
  • Figure 5: The cylinder $N$ in $T_xM$ containing the tetrahedron $\tilde{\Delta}$.
  • ...and 4 more figures

Theorems & Definitions (91)

  • Theorem 1.1
  • Corollary 1.2
  • Corollary 1.3
  • Definition 3.1
  • Definition 3.2
  • Definition 3.3
  • Theorem 3.4: Theorem 2 from Breslin by Breslin
  • Theorem 3.5: Extension of Breslin's theorem
  • Theorem 4.1
  • Theorem 4.2
  • ...and 81 more