Hyperbolic Lattices Derived From Denser Tessellations

The paper’s aims are threefold. First, to identify the Fuchsian groups associated with the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\{4\lambda, 4\}$ </tex-math></inline-formula> and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\{6\lambda, 3\}$ </tex-math></inline-formula> hyperbolic tessellations in quaternion orders, where <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\lambda = 3^{n}$ </tex-math></inline-formula>. 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 <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\Gamma _{4\lambda }$ </tex-math></inline-formula> and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\Gamma _{6\lambda }$ </tex-math></inline-formula> derived from quaternion algebras <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathcal {A}$ </tex-math></inline-formula> over a number field <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathbb {K}$ </tex-math></inline-formula> whose primitive elements are established in <xref ref-type="theorem" rid="theorem10" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Theorems 10</xref> and <xref ref-type="theorem" rid="theorem20" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">20</xref>, respectively. Consequently, <xref ref-type="theorem" rid="theorem13" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Theorems 13</xref> and <xref ref-type="theorem" rid="theorem14" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">14</xref>, and <xref ref-type="theorem" rid="theorem23" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Theorems 23</xref> and <xref ref-type="theorem" rid="theorem24" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">24</xref> establish the bases of the maximal orders related to the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\{4\lambda, 4\}$ </tex-math></inline-formula> and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\{6\lambda,3\}$ </tex-math></inline-formula> tessellations, respectively, leading to the algebraic labeling to be complete. Therefore, these results provide the algebraic structures for constructing the corresponding hyperbolic lattices.