Regular Tessellations of the Hyperbolic Plane by Fundamental Domains of a Fuchsian Group

For positive integers p and q with 1/p+1/q < 1/2, a tessellation of type {p, q} is a tessellation of the hyperbolic plane by regular p-gons with q p-gons meeting at each vertex. In this paper, a necessary and sufficient condition on the integers p and q is established to determine when a tessellation of type {p, q} can be realized as a tessellation of the hyperbolic plane by fundamental domains of some Fuchsian group. Specifically, a tessellation of type {p, q} is a tessellation by fundamental domains if and only if q has a prime divisor less than or equal to p. 2000 Math. Subj. Class. 20H10.