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The arc complexes of partially decorated hyperbolic polygons

Pallavi Panda

Abstract

We consider two families of hyperbolic polygons: ideal and ideal once-punctured, some of whose spikes are decorated with horoballs. We show that the arc complexes of these two families of surfaces, generated by edge-to-edge arcs and edge-to-decorated-spike arcs, are closed piecewise linear balls. This is proved in a completely combinatorial setting: compact polygons whose vertices are assigned red or blue colouring. In order to prove the ballness, we show that these simplicial complexes are pseudo-manifolds and use shellability to conclude. As a consequence, we parametrise weakly-lengthening deformations of the partially decorated hyperbolic polygons.

The arc complexes of partially decorated hyperbolic polygons

Abstract

We consider two families of hyperbolic polygons: ideal and ideal once-punctured, some of whose spikes are decorated with horoballs. We show that the arc complexes of these two families of surfaces, generated by edge-to-edge arcs and edge-to-decorated-spike arcs, are closed piecewise linear balls. This is proved in a completely combinatorial setting: compact polygons whose vertices are assigned red or blue colouring. In order to prove the ballness, we show that these simplicial complexes are pseudo-manifolds and use shellability to conclude. As a consequence, we parametrise weakly-lengthening deformations of the partially decorated hyperbolic polygons.
Paper Structure (23 sections, 32 theorems, 14 equations, 15 figures)

This paper contains 23 sections, 32 theorems, 14 equations, 15 figures.

Table of Contents

  1. Abstract.
  2. Introduction
  3. Organisation of the paper.
  4. Acknowledgements.
  5. Setup
  6. Simplicial complex
  7. Arcs and arc complexes
  8. Bicolourings.
  9. Shelling of finite arc complexes
  10. Main theorems
  11. Bicoloured convex polygon
  12. Punctured bicoloured polygons
  13. Applications: decorated hyperbolic polygons
  14. Hyperbolic geometry
  15. Horoball connection.
  16. ...and 8 more sections

Key Result

Theorem A

For $n\geq 3$ and $2\leq r\leq n$, the arc complex $\mathcal{A}\left(\Pi\right)$ of a partially decorated ideal polygon $\Pi\in\widehat{\Pi}_{n,r}$ is $PL$-homeomorphic to a closed ball of dimension $n+r-4$.

Figures (15)

  • Figure 1:
  • Figure 2: The three types of diagonals and the full arc complex of $\mathcal{P}_{4}^{\times}$
  • Figure 3:
  • Figure 4:
  • Figure 5:
  • ...and 10 more figures

Theorems & Definitions (45)

  • Theorem A
  • Theorem B
  • Theorem C
  • Theorem D
  • Theorem E
  • Lemma 2.1
  • Theorem 2.2
  • Definition 2.3
  • Theorem 2.4
  • Theorem 2.5
  • ...and 35 more