An algorithm to construct arithmetic Fuchsian groups derived from quaternion algebras and the corresponding hyperbolic lattices

The aim of this paper is to propose an algorithm to construct arithmetic Fuchsian groups derived from quaternion algebras and quaternion orders which will lead to the construction of hyperbolic lattices. To achieve this goal a necessary condition for obtaining arithmetic Fuchsian groups ΓpΓp from a tessellation {p,q}{p,q} whose regular hyperbolic polygon PpPp generates an oriented surface with genus g≥2g≥2 is established. This necessary condition is called Fermat condition due to its identification with the Fermat primes. It is also shown an isomorphism between arithmetic Fuchsian groups derived from different edge-pairings sets of the regular fundamental region associated with the tessellation {4g,4g}{4g,4g} for g=2n,3.2n,5.2n,and3.5.2n, and the tessellation {4g+2,2g+1}{4g+2,2g+1} for g=2g=2. One set uses the normal form whereas the other one uses diametrically opposite edge-pairings. All these transformations are hyperbolic and so result in an oriented compact Riemann surface.