Flip-graphs of 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 $\mathcal{M}\mathcal{F}({\Sigma }_n)$. Bounds on the diameter of $\mathcal{M}\mathcal{F}({\Sigma }_n)$ 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 Möbius strip, $\mathcal{M}\mathcal{F}({\Sigma }_n)$ 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.

arXiv comment37 pages, 17 figures