In this paper, we construct signal constellations from lattices in complex hyperbolic spaces. To construct a hyperbolic lattice, we identify an arithmetic Fuchsian group with the group of units O1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {O}<^>{1}$$\end{document} of a natural quaternion order O subset of V\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {O}\subset \mathcal {V}$$\end{document}, in which V\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {V}$$\end{document} is some quaternion algebra over an algebraic number field. The arithmetic Fuchsian group, G, is isomorphic to the fundamental group of a regular hyperbolic polygon P, and an oriented compact surface arises from the pairwise identification of its opposite edges. The polygon P is the fundamental region associated with a regular tessellation {p,q}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{p,q\}$$\end{document}. The main contribution of this paper is to employ the Reidemeister-Schreier rewriting process to use proper decompositions of the full symmetry group of a tessellation {p,q}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{p,q\}$$\end{document}, which allows the matching of uniform signal constellations to quotient groups of G. In this direction, we consider the labelling of phase-shift keying (PSK) and amplitude-phase keying (APK) signal constellations diagrams via three approaches: cyclic quotient groups, a direct product of cyclic quotient groups and semi-direct product of cyclic groups (the dihedral group case).
