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Polyhedral realisations of finite arc complexes using strip deformations

François Guéritaud, Pallavi Panda

TL;DR

This work establishes a polyhedral realization of the weakly admissible deformation cone for decorated hyperbolic surfaces with finite arc complexes. By introducing strip deformations along arcs and a strip map that identifies arc complexes with the projectivised deformation space, the authors show that $aroldsymbol{mbda}(m)^+$ is a finite polytope homeomorphic to the arc complex $oldsymbol{}( Pi)$, with faces indexed by spread subsets. They give a precise combinatorial description of facets and vertices, including detailed treatments of fully decorated cases (ideal polygons, punctured polygons, crowns, and Möbius strips), and provide explicit exception lists for small combinatorial types. This bridges hyperbolic geometry, arc-complex topology, and polyhedral combinatorics, yielding a toolkit to study infinitesimal deformations via combinatorial data and potentially informing related Lorentzian geometry constructions. The results give a concrete, computable realization of deformation cones as arc-complex-based polytopes, enabling explicit description of faces, dimensions, and vertices in the finite-arc-complex regime.

Abstract

We study infinitesimal deformations of complete hyperbolic surfaces with boundary and with ideal vertices, possibly decorated with horoballs. ``Admissible'' deformations are the ones that pull all horoballs apart; they form a convex cone of deformations. We describe this cone in terms of the arc complex of the surface: specifically, this paper focuses on the surfaces for which that complex is finite. Those surfaces form four families: (ideal) polygons, once-punctured polygons, one-holed polygons (or ``crowns''), and Möbius strips with spikes. In each case, we describe a natural simplicial decomposition of the projectivised admissible cone and of each of its faces, realizing them as appropriate arc complexes.

Polyhedral realisations of finite arc complexes using strip deformations

TL;DR

This work establishes a polyhedral realization of the weakly admissible deformation cone for decorated hyperbolic surfaces with finite arc complexes. By introducing strip deformations along arcs and a strip map that identifies arc complexes with the projectivised deformation space, the authors show that is a finite polytope homeomorphic to the arc complex , with faces indexed by spread subsets. They give a precise combinatorial description of facets and vertices, including detailed treatments of fully decorated cases (ideal polygons, punctured polygons, crowns, and Möbius strips), and provide explicit exception lists for small combinatorial types. This bridges hyperbolic geometry, arc-complex topology, and polyhedral combinatorics, yielding a toolkit to study infinitesimal deformations via combinatorial data and potentially informing related Lorentzian geometry constructions. The results give a concrete, computable realization of deformation cones as arc-complex-based polytopes, enabling explicit description of faces, dimensions, and vertices in the finite-arc-complex regime.

Abstract

We study infinitesimal deformations of complete hyperbolic surfaces with boundary and with ideal vertices, possibly decorated with horoballs. ``Admissible'' deformations are the ones that pull all horoballs apart; they form a convex cone of deformations. We describe this cone in terms of the arc complex of the surface: specifically, this paper focuses on the surfaces for which that complex is finite. Those surfaces form four families: (ideal) polygons, once-punctured polygons, one-holed polygons (or ``crowns''), and Möbius strips with spikes. In each case, we describe a natural simplicial decomposition of the projectivised admissible cone and of each of its faces, realizing them as appropriate arc complexes.
Paper Structure (35 sections, 27 theorems, 31 equations, 18 figures)

This paper contains 35 sections, 27 theorems, 31 equations, 18 figures.

Table of Contents

  1. Introduction
  2. Definitions
  3. Topological type and dimension
  4. Notation
  5. Ideal polygons.
  6. Once-punctured polygons.
  7. Crowns.
  8. Möbius strip with spikes.
  9. Admissible deformations
  10. Horoball connections
  11. The admissible cone
  12. Subsurfaces
  13. Numbers of horoball connections and closed curves
  14. Arc complexes
  15. Strip deformations
  16. ...and 20 more sections

Key Result

Theorem 1.1

Let $\Pi$ be a (decorated) surface with finite arc complex, and $m\in \mathfrak{D}(\Pi)$ a hyperbolic metric. There is a natural parametrization of the closed polyhedron of (weakly) admissible deformations $\overline\Lambda(m)^+$ in the positive projectivization $\mathbb{P}^+ \mathrm{T}_m\mathfrak{D

Figures (18)

  • Figure 1: The four families of surfaces with finite arc complexes. The index $n$ refers to the number of spikes: here $\Pi^\Diamond_6, \Pi^\times_3, \Pi^\circledcirc_3~, \mathcal{M}_1$.
  • Figure 2: Left panel: A spread subset (red with blue interior) in a partially decorated crown. Right panel: Not a spread subset because $\Pi\smallsetminus P$ has a component that does not meet $\partial \Pi$.
  • Figure 3: Decomposition of the four surfaces by a horoball connection $\beta$.
  • Figure 5: Decomposition of a Möbius strip by a non-separating horoball connection $\beta$.
  • Figure 6: Left-panel: Any horoball connection disjoint from an edge-to-edge arc is disjoint from a spike-to-edge arc. Right panel: The spike-to-edge arc $\alpha$ is the only arc disjoint from the horoball connections $\beta_1,\ldots, \beta_4$.
  • ...and 13 more figures

Theorems & Definitions (62)

  • Theorem 1.1
  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Remark 2.4
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • Definition 2.8
  • Definition 2.10
  • ...and 52 more