Quadrangulations of sphere and ball quotients

Abstract

We give a classification of sphere quadrangulations satisfying a condition of non‐negative curvature, following Thurston's classification of sphere triangulations under the same condition. The generic family of quadrangulations is parametrized by the points of positive square‐norm of an integral Gaußian lattice Λ′ in the six‐dimensional complex Lorentz space. There is a subgroup of automorphisms of Λ′ acting on this lattice whose orbits parametrize sphere quadrangulations in a one‐to‐one manner. This group acts discretely on the corresponding five‐dimensional complex hyperbolic space; is of finite co‐volume; its ball quotient is the moduli space of unordered 8 points on the Riemann sphere, and also appears in Picard‐Terada‐Deligne‐Mostow list. Both Thurston's lattice and our lattice may be thought of as parametrizations of certain families of subgroups of the modular group; equivalently, of certain families of dessins. These families also parametrize points of a moduli space.

Complete flat cone metrics on punctured surfaces

“…In addition, regarding combinatorial triangulations or quadrangulations as cone metrics as in [12], one can parametrize certain families of dessins d'enfants. See [19], [17], [20], [21], [1].…”

Which Cubic Graphs have Quadrangulated Spherical Immersions?

“…Zeytin and Uludag show that for each possible triple (k 1 , k 2 , k 3 ), where k i is the quantity of vertices of degree i (the number of vertices of degree 4 is unlimited), there is a lattice Γ such that the orbits under a certain group of automorphisms of Γ give a one-to-one parametrization of quadrangulations corresponding to the given degree data. For the case (0, 0, 8), which is exactly the case of interest to us, the quotient space corresponds to the moduli space of 8 unordered points on the Riemann sphere [21].…”