Hyperbolic Lattices Derived From Denser Tessellations

The paper’s aims are threefold. First, to identify the Fuchsian groups associated with the $\{4\lambda, 4\}$ and $\{6\lambda, 3\}$ hyperbolic tessellations in quaternion orders, where $\lambda = 3^{n}$ . Second, to provide the bases of the maximal quaternion orders associated with such dense tessellation families. Third, to provide the algebraic structure for constructing hyperbolic lattices or geometrically uniform signal constellations leading to a complete algebraic labeling. These tessellations are relevant for being denser and having inherent mathematical properties, making them a favorite in several classical and quantum applications. However, to the best of our knowledge, a systematic approach and results are needed to put the applications of such relevant mathematical structures in perspective. Hence, the contributions of this paper are related to the presentation of the corresponding Fuchsian groups $\Gamma _{4\lambda }$ and $\Gamma _{6\lambda }$ derived from quaternion algebras $\mathcal {A}$ over a number field $\mathbb {K}$ whose primitive elements are established in Theorems 10 and 20, respectively. Consequently, Theorems 13 and 14, and Theorems 23 and 24 establish the bases of the maximal orders related to the $\{4\lambda, 4\}$ and $\{6\lambda,3\}$ tessellations, respectively, leading to the algebraic labeling to be complete. Therefore, these results provide the algebraic structures for constructing the corresponding hyperbolic lattices.