Hurwitz theory of elliptic orbifolds, I

An elliptic orbifold is the quotient of an curve by a finite group. Eskin and Okounkov proved that generating functions for number branched covers with specified ramification are quasimodular forms full modular group $SL_2(\mathbb{Z})$. They later generalized this theorem to enumeration pillowcase, i.e. involution, proving quasi-modularity $\Gamma_1(2)$. We generalize their work cyclic groups orders $N=3$, $4$, $6$, level $\Gamma_1(N)$. One corollary certain hexagon, square, triangle tilings compact surfaces quasi-modular. These enumerate lattice points in moduli spaces flat surfaces. analyze asymptotic behavior as tiles goes infinity, theoretically giving algorithm compute Masur-Veech volumes cubic, quartic, sextic differentials. also deduce volume polynomial $\pi$.