Flip-graphs of non-orientable filling surfaces

TL;DR

This paper investigates the flip-graph of non-orientable filling surfaces, establishing bounds on its diameter growth rate and providing exact asymptotics for the Möbius strip case, extending known results from orientable surfaces.

Contribution

It extends diameter bounds of flip-graphs to non-orientable surfaces and determines exact growth rate for the Möbius strip case.

Findings

Diameter grows at least like 5n/2 and at most like 4n for non-orientable surfaces.

Exact diameter growth rate of 5n/2 for the Möbius strip case.

Provides bounds on flip-graph diameters for non-orientable filling surfaces.

Abstract

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