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 \(\mathcal {O}^{1}\) of a natural quaternion order \(\mathcal {O}\subset \mathcal {V}\) , in which \(\mathcal {V}\) 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 \(\{p,q\}\) . 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 \(\{p,q\}\) , 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).
