Triangulations of the Sphere

Abstract

Thurston gave a simple way to construct all triangulations of the sphere for which 5 or 6 triangles meet at each vertex, using the Eisenstein integers $\mathbb{E}$. While such triangulations can be defined purely combinatorially, Thurston noticed that given such a triangulation, one can make all the triangles into flat equilateral triangles with the same edge length, and this gives the 2-sphere a flat Riemannian metric except at 12 cone points with angle deficit $\ddot{I}€/3$. He showed that up to rescaling, all such Riemannian metrics arise from his procedure. He studied the moduli space $\mathcal{M}$ of all such metrics modulo rescaling, and showed that $\mathcal{M}$ is open and dense in an orbifold $\overline{\mathcal{M}}={\mathbb{PC}}_{+}^{10}/\hat{I}“$. Here ${\mathbb{C}}_{+}^{10}=\{v\in {\mathbb{C}}^{10}|Q(v)>0\}$ for some quadratic form $Q$ of signature $(1,9)$ on ${\mathbb{C}}^{10}$, ${\mathbb{PC}}_{+}^{10}$ is its projectivization, and $\hat{I}“$ is a certain discrete group of linear transformations of ${\mathbb{C}}^{10}$ preserving both $Q$ and the lattice ${\mathbb{E}}^{10}\subset {\mathbb{C}}^{10}$. He also showed that $\overline{\mathcal{M}}$ is the moduli space of flat Riemannian metrics on the sphere with at most $12$ cone points and angle deficits that are positive integer multiples of $\ddot{I}€/3$. Here we briefly outline the basic ideas behind this work, and illustrate them with examples.

3 pages, expanded and corrected version of the published article