Secondary Fans and Secondary Polyhedra of Punctured Riemann Surfaces

Abstract

A famous construction of Gel'fand, 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.

Keywords:

• ideal triangulations
• decorated Teichmüller space
• ideal polytope

MATHEMATICS SUBJECT CLASSIFICATION:

• 30F60 (32G15, 52B12, 57M50)

Additional information

Funding

This work is supported by DFG via SFB-TRR 109: “Discretization in Geometry and Dynamics.” Further support for M. Joswig from Einstein Foundation Berlin (within the framework of Matheon) and DFG via SFB-TRR 195: “Symbolic Tools in Mathematics and their Application.”