Complete hyperbolic lattices derived from tessellations of type {4g, 4g}

Regular tessellations of the hyperbolic plane play an important role in the design of signal constellations for digital communication systems. Self-dual tessellations of type {4g, 4g} with g = 2(n), 3 . 2(n), and 5 . 2(n) have been considered where the corresponding arithmetic Fuchsian groups are derived from quaternion orders over quadratic extensions of the rational. The objectives of this work are to establish the maximal orders derived from {4g, 4g} tessellations for which the hyperbolic lattices are complete (the motivation for constructing complete hyperbolic lattices is their application to the design of hyperbolic lattice codes), and to identify the arithmetic Fuchsian group associated with a quaternion algebra and a quaternion order.