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Bi-geodesic mappings between hyperbolic surfaces with boundary

Wen Yang

TL;DR

The paper proves rigidity for bi-geodesic mappings between interiors of compact hyperbolic surfaces with boundary: any bijection $f$ that and its inverse map every geodesic to a geodesic must be an isometry. The authors develop a framework based on preserved geodesic types and the arc-and-curve complex $AC(S)$, showing $f$ induces a topological type $h$ via a realization of $AC(S)$, and then upgrade a corresponding homeomorphism to an isometry through a rigidity argument. The approach combines geodesic classifications (simple closed, boundary-associating, spiraling), pants decompositions, and geodesic elementary moves to transfer combinatorial data to geometric rigidity. This yields an isometry classification of bi-geodesic mappings for surfaces with boundary and suggests avenues for generalization and open problems for cases with no boundary.

Abstract

It is proved that a bijection between two compact hyperbolic surfaces with boundary is an isometry if it and its inverse map each geodesic onto some geodesic.

Bi-geodesic mappings between hyperbolic surfaces with boundary

TL;DR

The paper proves rigidity for bi-geodesic mappings between interiors of compact hyperbolic surfaces with boundary: any bijection that and its inverse map every geodesic to a geodesic must be an isometry. The authors develop a framework based on preserved geodesic types and the arc-and-curve complex , showing induces a topological type via a realization of , and then upgrade a corresponding homeomorphism to an isometry through a rigidity argument. The approach combines geodesic classifications (simple closed, boundary-associating, spiraling), pants decompositions, and geodesic elementary moves to transfer combinatorial data to geometric rigidity. This yields an isometry classification of bi-geodesic mappings for surfaces with boundary and suggests avenues for generalization and open problems for cases with no boundary.

Abstract

It is proved that a bijection between two compact hyperbolic surfaces with boundary is an isometry if it and its inverse map each geodesic onto some geodesic.

Paper Structure

This paper contains 14 sections, 26 theorems, 13 equations, 17 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Hyperbolic surfaces and geodesics
  4. Arc and curve complex
  5. Elementary move
  6. Outline of proof
  7. A preliminary classification of geodesics
  8. Topological type of bi-geodesic mapping
  9. A classification of associations
  10. A realization of arc and curve complex
  11. Homotopy classes of simple arcs
  12. From bijection to isometry
  13. Proof of main theorem
  14. Concluding Remarks

Key Result

Theorem 2

Let $R_{i}$ be a compact hyperbolic surface of genus $g_{i}$ with $n_{i}\geqslant1$ geodesic boundary components, and $S_{i}$ the interior of $R_{i}$, $i=1,2$. Suppose that $f:S_{1}\rightarrow S_{2}$ is a bijection such that $f$ and $f^{-1}$ map each geodesic onto some geodesic. Then $f$ is an isome

Figures (17)

  • Figure 1: A non-trivial bi-geodesic mapping.
  • Figure 2: If $\eta$ is a geodesic with self-intersection, then there exists another lift $\tilde{\alpha}_{2}'\cup\tilde{\beta}_{2}\cup\tilde{\xi}_{2}\cup\tilde{\eta}_{2}$ of $\alpha'\cup\beta\cup\xi\cup\eta$ such that $\tilde{\eta}$ intersects $\tilde{\eta}_{2}$. Now $\tilde{\xi}$ must intersect $\tilde{\xi}_{2}$, which contradicts the fact that $\xi$ is simple.
  • Figure 3: $\tau_{k}\rightarrow\rho$.
  • Figure 4: A simple self-intersection.
  • Figure 5: Construction of $\tilde{\rho}$.
  • ...and 12 more figures

Theorems & Definitions (42)

  • Example 1
  • Theorem 2
  • Corollary 3
  • Proposition 4: liu2019geodesic
  • Lemma 5: Collar lemma, see Lemma 13.6 in farb2011primer
  • Lemma 6: hubbard2006teichmuller
  • Theorem 7: liu2019geodesic
  • Proposition 8
  • Theorem 9: Gauss-Bonnet
  • Theorem 10: see korkmaz2010arc
  • ...and 32 more