Flip-graph moduli spaces of filling surfaces

Abstract

This paper is about the geometry of the flip-graphs associated to triangulations of surfaces. More precisely, we consider a topological surface with a privileged boundary curve and study the space of its triangulations with n vertices on this 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 these graphs up to the action of the mapping class group. This always results in finite graphs, which we call modular flip-graphs. Our main focus is on the diameter growth of these graphs as n increases. We obtain general estimates that hold for filling surfaces of any topological type. 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.