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Triangulating surfaces quasi-isometrically

Agelos Georgakopoulos, Federico Vigolo

Abstract

We prove that every complete Riemannian surface $(Σ,d_Σ)$ admits a triangulation $D$ whose 1-skeleton, when endowed with the inherited length metric, is quasi-isometric to $(Σ,d_Σ)$. Moreover, the faces of $D$ have intrinsic diameters uniformly bounded by an arbitrarily small constant. We also prove that $(Σ,d_Σ)$ admits a uniform net if and only if it has a triangulation whose 1-skeleton is quasi-isometric to it when equipped with the simplicial metric.

Triangulating surfaces quasi-isometrically

Abstract

We prove that every complete Riemannian surface admits a triangulation whose 1-skeleton, when endowed with the inherited length metric, is quasi-isometric to . Moreover, the faces of have intrinsic diameters uniformly bounded by an arbitrarily small constant. We also prove that admits a uniform net if and only if it has a triangulation whose 1-skeleton is quasi-isometric to it when equipped with the simplicial metric.
Paper Structure (22 sections, 22 theorems, 16 equations, 6 figures)

This paper contains 22 sections, 22 theorems, 16 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Generalising the setup
  3. Eliminating the multiplicative or additive distortion
  4. On mapping class groups and flip graphs.
  5. Notation and preliminaries
  6. Quasi-isometries and nets
  7. Curves
  8. Riemannian surfaces
  9. Graphs and triangulations
  10. Resolving intersections of curves
  11. Arcs in minimal positions and $\epsilon$-geodesics
  12. The main result
  13. Constructing a quasi-isometrically embedded metric graph
  14. Bounding the extrinsic diameter of the faces
  15. Proving that faces are discs
  16. ...and 7 more sections

Key Result

Theorem 1

Let $(\Sigma,d_\Sigma)$ be a complete Riemannian surface, and $\Xi\in \mathbb R_{\geq 0}$. Then there is a triangulation $\mathcal{D}$ of $(\Sigma,d_\Sigma)$ each 2-cell $C$ of which has diameter at most $\Xi$ with respect to its length metric $d^C_\ell$. Moreover, the identity map from the 1-skelet

Figures (6)

  • Figure 1: Transforming a rooted tree $T$ into a collection of disjoint paths meeting at the root (blue lines in the picture).
  • Figure 2: Constructing nested surfaces around the middle portion of a very long curve $c$.
  • Figure 3: Constructing a curve $\gamma$ with endpoints in $X$ that cut $c$ in the case that $N_1\smallsetminus c_1$ is disconnected.
  • Figure 4: The thick lines are disjoint $\epsilon$-geodesics. In the case where the curves crossing the short path $\beta_m$ are not connected, removing them and adding $\epsilon$-geodesics from their endpoints to $v_m$ increases the cardinality of the family
  • Figure 5: The thick lines are disjoint $\epsilon$-geodesics. If there is a point $m$ connected to $\partial P$ via two short paths $\beta_m^\pm$ that intersect $\epsilon$-geodesics connected to the starting curve $\alpha_0$, we increase the number of disjoint $\epsilon$-geodesic by removing one of them and adding $\epsilon$-geodesics from the endpoints of the other one.
  • ...and 1 more figures

Theorems & Definitions (48)

  • Theorem 1
  • Theorem 2
  • Corollary 3
  • Remark 1.1
  • Corollary 4
  • Lemma 2.2
  • proof : Sketch of proof
  • Remark 2.3
  • Remark 2.4
  • Remark 2.5
  • ...and 38 more