Secondary fans of punctured Riemann surfaces

Abstract:

A famous construction of Gelfand, Kapranov and Zelevinsky associates to each finite point configuration \(A \subset \mathbb{R}^d \) a polyhedral fan, which stratifies the space of weight vectors by the combinatorial types of regular subdivisions of \(A\). That fan arises as the normal fan of a convex polytope. In a completely analogous way we associate to each hyperbolic Riemann surface \(R\) with punctures a polyhedral fan. Its cones correspond to the ideal cell decompositions of \(R\) that occur as the horocyclic Delaunay decompositions which arise via the convex hull construction of Epstein and Penner. Similar to the classical case, this secondary fan of \(R\) turns out to be the normal fan of a convex polyhedron, the secondary polyhedron of \(R\). This is joint work with Michael Joswig and Boris Springborn.