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Flip-graphs of non-orientable filling surfaces

Pallavi Panda, Hugo Parlier, Lionel Pournin

TL;DR

The paper extends the theory of flip-graphs and their diameters from orientable to non-orientable filling surfaces, establishing that the modular flip-graph MF(Σ_n) has linear diameter with asymptotic constants c_Σ satisfying 5/2 ≤ c_Σ ≤ 4 for non-orientable Σ, and showing the Möbius strip attains the lower bound with diameters growing like 5n/2. It develops strong convexity results for arcs parallel to the privileged boundary in the non-orientable setting, and provides both lower- and upper-bounding techniques: explicit constructions (A_n^−, A_n^+) to prove lower bounds, and a central-triangle method plus canonical target triangulations to prove upper bounds. A detailed study of simplicial triangulations yields bounds on the diameter of F_⋆(M_n), which, together with a translation to F(M_n), gives near-tight control over the diameter growth and confirms the asymptotic 5n/2 rate in key non-orientable cases. Overall, the work advances understanding of mapping-class-type geometry via modular flip-graphs in the non-orientable regime, including precise asymptotics for the Möbius strip and explicit diameter bounds for the non-simplicial triangulation space.

Abstract

Consider a surface $Σ$ with punctures that serve as marked points and at least one marked point on each boundary component. We build a filling surface $Σ_n$ by singling out one of the boundary components and denoting by $n$ the number of marked points it contains. We consider the triangulations of $Σ_n$ whose vertices are the marked points and the associated flip-graph $\mathcal{F}(Σ_n)$. Quotienting $\mathcal{F}(Σ_n)$ by the homeomorphisms of $Σ$ that fix the privileged boundary component results in a finite graph $\mathcal{MF}(Σ_n)$. Bounds on the diameter of $\mathcal{MF}(Σ_n)$ are available when $Σ$ is orientable and we provide corresponding bounds when $Σ$ is non-orientable. We show that the diameter of this graph grows at least like $5n/2$ and at most like $4n$ as $n$ goes to infinity. If $Σ$ is an unpunctured Möbius strip, $\mathcal{MF}(Σ_n)$ coincides with $\mathcal{F}(Σ_n)$ and we prove that the diameter of this graph grows exactly like $5n/2$ as $n$ goes to infinity.

Flip-graphs of non-orientable filling surfaces

TL;DR

The paper extends the theory of flip-graphs and their diameters from orientable to non-orientable filling surfaces, establishing that the modular flip-graph MF(Σ_n) has linear diameter with asymptotic constants c_Σ satisfying 5/2 ≤ c_Σ ≤ 4 for non-orientable Σ, and showing the Möbius strip attains the lower bound with diameters growing like 5n/2. It develops strong convexity results for arcs parallel to the privileged boundary in the non-orientable setting, and provides both lower- and upper-bounding techniques: explicit constructions (A_n^−, A_n^+) to prove lower bounds, and a central-triangle method plus canonical target triangulations to prove upper bounds. A detailed study of simplicial triangulations yields bounds on the diameter of F_⋆(M_n), which, together with a translation to F(M_n), gives near-tight control over the diameter growth and confirms the asymptotic 5n/2 rate in key non-orientable cases. Overall, the work advances understanding of mapping-class-type geometry via modular flip-graphs in the non-orientable regime, including precise asymptotics for the Möbius strip and explicit diameter bounds for the non-simplicial triangulation space.

Abstract

Consider a surface with punctures that serve as marked points and at least one marked point on each boundary component. We build a filling surface by singling out one of the boundary components and denoting by the number of marked points it contains. We consider the triangulations of whose vertices are the marked points and the associated flip-graph . Quotienting by the homeomorphisms of that fix the privileged boundary component results in a finite graph . Bounds on the diameter of are available when is orientable and we provide corresponding bounds when is non-orientable. We show that the diameter of this graph grows at least like and at most like as goes to infinity. If is an unpunctured Möbius strip, coincides with and we prove that the diameter of this graph grows exactly like as goes to infinity.
Paper Structure (5 sections, 32 theorems, 44 equations, 10 figures)

This paper contains 5 sections, 32 theorems, 44 equations, 10 figures.

Table of Contents

  1. Introduction
  2. From orientable to non-orientable surfaces
  3. Far apart triangulations of non-orientable filling surfaces
  4. An upper bound on the diameter of $\mathcal{F}(\mathrm{M}_n)$
  5. Bounds on the diameter of $\mathcal{F}_\star(\mathrm{M}_n)$

Key Result

Theorem 1.1

If $\Sigma$ is a non-orientable filling surface, then Moreover, if $\Sigma$ is a demigenus $g$, non-orientable one-holed surface, then when $g$ is at least $3$ and $c_\Sigma$ is at most $23/8$ when $g$ is equal to $2$.

Figures (10)

  • Figure 1: A triangulation of the Möbius strip with two marked points in the boundary (shown in the cross-cap model of the Möbius strip) and a triangulation of the once-punctured disk with two marked points in the boundary.
  • Figure 2: The contraction of $\alpha$.
  • Figure 3: If $\Sigma$ is not one holed we cut it into $\Sigma^\otimes$ and $\Sigma^\odot$.
  • Figure 4: The triangulations $A_n^-$ (first two rows) and $A_n^+$ (third row) when $n$ is even (left) and odd (right). The first row shows $A_n^-$ when $\Sigma$ is one-holed, and the second when it is not.
  • Figure 7: The two flips incident to arc $\alpha$ along the path considered in the proof of Proposition \ref{['NOFG.sec.3.lem.3']}.
  • ...and 5 more figures

Theorems & Definitions (35)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 2.1
  • Theorem 2.2
  • Proposition 2.3
  • Theorem 2.4: ParlierPournin2017
  • Theorem 2.5
  • Lemma 2.6
  • Theorem 2.7
  • ...and 25 more