# On Thurston's triangulations of sphere

I have two questions from Thurston's paper [1].

In the paper [1], Thurston talks about classifying certain classes of triangulations of the sphere. Here a triangulation of a sphere a Topological sphere that is formed by glueing together equilateral triangles. Two triangulations are isomorphic if there is a metric space isometry between the two spheres that are also orientation preserving, maps vertices to vertices and edges to edges. Thurston studies about a special class of triangulations, $$P(n; k_1, \cdots, k_r)$$ where there are $$r$$ vertices (called cone points) with $$6 - k_i$$ number of triangles around them and the rest with exactly $$6$$ triangles.

Given $$T \in P(n; k_1, \cdots, k_r)$$, we'll define the developing map $$D_T \colon \widetilde{T} \rightarrow \mathbb{C}$$. Where $$\widetilde{T}$$ is the universal cover of $$T - \{x_1, \cdots, x_r\}$$ where $$x_i$$ is a singular point. The developing map maps one triangle to the triangle $$(0, 1, \omega)$$ in $$\mathbb{C}$$ where $$\omega$$ is the 6th root of unity; the rest of the map is an analytic continuation.

Here's the part that is hard to understand (pp. 8/ pp. 518):

"A particularly interesting phenomenon happens when the number of triangles around a singular vertex is $$1$$, $$2$$ or $$3$$. Consider the component $$N_v$$ of the inverse image in $$\widetilde{T}$$ of a small neighborhood of any such vertex $$v$$. It develops into the vicinity of some vertex $$w$$ in Eis (Eis is the set $$a + b\omega$$ for $$a, b \in \mathbb{Z}$$ and $$\omega$$ the 6th root of unity) In these cases, the number of triangles around $$v$$ is a divisor of $$6$$, so the developing map repeats itself when it first wraps around the vertex $$w$$, along a path in $$\widetilde{T}$$ which maps to a curve in $$T$$ wrapping respectively $$6, 3$$ or $$2$$ times around $$v$$. Therefore the developing map is defined from a smaller covering of $$T$$ minus its singular vertices which can be obtained as a certain quotient space $$S(T)$$ of $$\widetilde{T}$$. In $$S(T)$$, each component of the preimage of a small neighborhood of $$v$$ only intersects in six triangles. In fact $$S(T)$$ is isomorphic to $$\mathbb{C}$$. Therefore $$T$$ is a quotient space of a discrete group $$\Gamma(T)$$ acting on $$\mathbb{C}$$ such that only elements of Eis are fixed points of elements of $$\Gamma(T)$$

For $$P(n; 4, 4, 4)$$ or $$P(n; 3, 4, 5)$$, the group $$\Gamma(T)$$ is a triangle group"

Questions:

1. How does Thurston go from "there exists a smaller cover" to "$$T$$ is a quotient of a discrete group"? Thurston is claiming that the space $$P(n; k_1, \cdots, k_r)$$ can be thought of as a quotient of $$\mathbb{C}$$.

2. For the specific examples with $$3$$ cone points, why is the group a triangle group?

[1] William P. Thurston, Shapes of polyhedra and triangulations of the sphere. https://arxiv.org/abs/math/9801088

• My comment doesn't make sense. I think I get it now: If $\Gamma$ is the group of deck transformations, then every element of $\mathbb{C}/\Gamma$ corresponds to a unique triangulation. This correspondence is also surjective. Thus $\mathbb C/\Gamma$ is the space of triangulations. The triangle group part follows from the fact that the fundamental group of a thrice punctured sphere is a triangle group. Aug 24, 2019 at 14:47