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

TL;DR

This paper establishes a precise criterion involving prime divisors for when regular hyperbolic tessellations of type {p,q} can be realized as fundamental domains of Fuchsian groups, linking geometric and algebraic properties.

Contribution

It provides a necessary and sufficient condition based on prime divisors for hyperbolic tessellations to be fundamental domains of Fuchsian groups.

Findings

A tessellation of type {p,q} can be realized as a Fuchsian group fundamental domain if and only if q has a prime divisor ≤ p.

The paper characterizes all such tessellations in terms of prime divisors of q.

It bridges geometric tessellations with algebraic properties of Fuchsian groups.

Abstract

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.