Secondary fans and secondary polyhedra of punctured Riemann surfaces

TL;DR

This work develops a GKZ-style polyhedral framework for punctured Riemann surfaces by translating Epstein–Penner's horocycle-based convex hull construction into a labeled, decorated setting. It defines a secondary fan $\Sigma(\mathcal{R})$ on cusp weights and proves it is the outer normal fan of a (generally unbounded) secondary polyhedron $\Sigma\text{-poly}(\mathcal{R},x)$ parameterized by a point $x\in\mathcal{R}$, with GKZ vectors $\phi_{x,T}$ encoding triangulations $T$. The authors highlight important departures from the classical GKZ theory: not all top-dimensional cones arise from triangulations, and adjacencies can involve multiple flips; the construction yields a polyhedron bundle over $\mathcal{R}$ and has practical, computable implications via a polymake-based workflow. The framework links decorated Teichmüller theory with polyhedral geometry, offering new tools for understanding ideal Delaunay decompositions and potential applications to discrete uniformization and hyperbolic geometry algorithms.

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 $R$.

Figures (12)

• Figure 1: Projections between the hyperboloid (${\mathbb H}$), Beltrami--Klein ($K$), hemisphere ($N$) and half-plane ($U$) models of the hyperbolic plane.
• Figure 2: Signed distances of horocycles and $\lambda$-lengths.
• Figure 3: Decorated (oriented) ideal triangle $\Delta=(a,b,c)$ in the upper half plane with $\lambda$- and $h$-lengths.
• Figure 4: Decorated ideal quadrilateral in the upper half-plane with $\lambda$- and $h$-lengths. The diagonal $e$ induces a triangulation with triangles $\Delta=(a,b,e)$ and $\Delta'=(c,d,e)$.
• Figure 5: Fuchsian group generated by the two parabolic transformations $A(z)=z+2$ and $B(z)=\tfrac{z}{2z+1}$ as a subgroup of $\mathop{\mathrm{PSL_2(\mathbb{R})}}\nolimits2R$ (left). The blue shaded redion is a fundamental domain. The resulting Riemann surface is a sphere with three cusps (right).

Theorems & Definitions (29)

• Lemma 2.1
• Lemma 2.2
• Remark 2.4
• Definition 2.5
• Theorem 2.6
• Remark 4.1
• Proposition 5.1: epstein_penner, pennerbook
• Definition 5.2
• Definition 5.3
• Lemma 5.4