Topological recursion of hyperbolic surfaces with tight boundaries


The Weil–Petersson volume of the moduli space of hyperbolic surfaces with geodesic boudaries is known to be given by a polynomial in the boundary lengths and to satisfy Mirzakhani’s topological recursion formula. I’ll discuss how to extend this to hyperbolic surfaces in which a subset of the boundaries is required to be tight, where we say a boundary is tight if it has minimal length among all curves that separate it from the other boundaries in the subset. The generating function of Weil–Petersson volumes of surfaces with fixed genus and fixed number of tight boundaries is again polynomial and satisfies a topological recursion that generalizes Mirzakhani’s formula. This joint work with Bart Zonneveld is largely inspired by recent works by Bouttier, Guitter & Miermont on the enumeration of planar maps with tight boundaries.

KdVI, University of Amsterdam