Secondary Fans And Secondary Polyhedra Of Punctured Riemann Surfaces

A famous construction of Gel'fand, Kapranov and Zelevinsky associates to each finite point configuration A subset of Rd 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 Script capital R with punctures a polyhedral fan. Its cones correspond to the ideal cell decompositions of Script capital 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 Script capital R turns out to be the normal fan of a convex polyhedron, the secondary polyhedron of Script capital R.