Quadrangulations of sphere and ball quotients

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 in the six-dimensional complex Lorentz space. There is a subgroup of automorphisms of 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. (C) 2013 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim