Regular tessellations of the hyperbolic plane by fundamental domains of a Fuchsian group

Understand

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.