# Tilings and Hurwitz Theory

# Tilings and Hurwitz Theory

Consider the tilings of an oriented surface by triangles, or squares, or hexagons, up to combinatorial equivalence. The *combinatorial curvature *of a vertex is 6, 4, or 3 minus the number of adjacent polygons, respectively.

Tilings are naturally stratified into all such having the same set of non-zero curvatures. We outline a proof that for squares and hexagons, the generating function for the number of tilings in a fixed stratum lies in a ring of quasi-modular forms of specified level and weight. Tiled surfaces are in bijection with certain ramified covers of *elliptic orbifolds, *quotients of elliptic curves by finite groups. Generalizing techniques pioneered by Eskin and Okounkov, who studied the elliptic curve and pillowcase orbifold, we express the generating function for a stratum in terms of the q-trace of an operator acting on Fock space, and recompute this trace in a new basis.

Quasi-modularity gives an algorithmic method to compute the Masur-Veech volumes of moduli spaces of cubic and quartic differentials, and implies these volumes are polynomial in pi.