Flip-graph moduli spaces of filling surfaces

TL;DR

This paper studies the geometric properties of flip-graph moduli spaces associated with triangulations of filling surfaces, focusing on diameter growth and providing exact results for specific surface types.

Contribution

It introduces a framework for analyzing flip-graph moduli spaces of filling surfaces and derives new diameter estimates, including exact diameters for certain cases.

Findings

General diameter growth estimates for all filling surfaces

Precise diameter estimates for specific families of filling surfaces

Exact diameter of modular flip-graphs for a cylinder with one vertex

Abstract

This paper is about the geometry of flip-graphs associated to triangulations of surfaces. More precisely, we consider a topological surface with a privileged boundary curve and study the spaces of its triangulations with n vertices on the boundary curve. The surfaces we consider topologically fill this boundary curve so we call them filling surfaces. The associated flip-graphs are infinite whenever the mapping class group of the surface (the group of self-homeomorphisms up to isotopy) is infinite, and we can obtain moduli spaces of flip-graphs by considering the flip-graphs up to the action of the mapping class group. This always results in finite graphs and we are interested in their geometry. Our main focus is on the diameter growth of these graphs as n increases. We obtain general estimates that hold for all topological types of filling surface. We find more precise estimates for certain families of filling surfaces and obtain asymptotic growth results for several of them. In particular, we find the exact diameter of modular flip-graphs when the filling surface is a cylinder with a single vertex on the non-privileged boundary curve.