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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 of a natural quaternion order O ⊂ V, in which 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 phaseshift 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). [ABSTRACT FROM AUTHOR]

This paper presents the conditions under which a coded digital communication system's performance may be improved using better signal sets design matched to groups. This improvement is directly related to the determination of the uniformization region of a hyperelliptic curve by using a Fuchsian differential equation and, consequently, the determination of the Fuchsian group and subgroup generators. The hyperelliptic curves of interest are given by y 2 = z n ± 1 , where n ∈ I N and n = 2 g + 1 or n = 2 g + 2 , and satisfy the following conditions: all the roots are distinct, they are symmetrically arranged (regular polygon) in the Poincaré disk, the juxtaposition of two such polygons gives the uniformization region. These conditions imply that the uniformization regions are regular polygons and that the associate Fuchsian group leads to regular tessellations from which arithmetic Fuchsian groups may be found. This work aims to answer and establish the conditions for the following questions: Can Fuchsian subgroups corresponding to two distinct hyperelliptic curves with the same degree and genus be isomorphic in D 2 as well as in H 2 ? What are the conditions under which an isometry acting on isomorphic Fuchsian subgroups in D 2 remain isomorphic in H 2 ? To show the isomorphism, we use the concept of conjugation. The first question's positive response is achieved for hyperelliptic curves given by y 2 = z n ± 1 . Finally, from the results established in D 2 and H 2 , if the Fuchsian subgroups are isomorphic by conjugation in D 2 , they remain isomorphic in H 2 by the action of an appropriate isometry as established in Theorem 3 and its generalization in Theorem 4. [ABSTRACT FROM AUTHOR]

We consider Riemann surfaces of even genus g with the action of the group D n × Z 2 , with n even. These surfaces have the maximal number of 4 non-conjugate symmetries and shall be called s-extremal. We show various results for such surfaces, concerning the total number of ovals, topological types of symmetries, hyperellipticity degree and the minimal genus problem. If in addition an s-extremal Riemann surface has the maximal total number of ovals, then it shall simply be called extremal. In the main result of the paper we find all the families of extremal Riemann surfaces of even genera, depending on if one of the symmetries is fixed-point free or not. [ABSTRACT FROM AUTHOR]

We show that any non-standard generating pair of a hyperbolic triangle group is represented by a special almost orbifold covering with a good marking. [ABSTRACT FROM AUTHOR]

In this paper, we first prove that for a Fuchsian group G of divergence type and non-lattice, if h is a quasi-symmetric homeomorphism of the real axis R corresponding to a quasi-conformal compact deformation of G, then h is not strongly singular for divergence groups. This generalizes a result of Bishop and Steger (1993). Furthermore, we show that Bishop and Steger's result does not hold for the covering groups of all d-dimensional 'Jungle Gyms' (d is any positive integer) which generalizes Gönye's results (2007) where the author discussed the case of 1-dimensional 'Jungle Gym'. [ABSTRACT FROM AUTHOR]

We show that if a cusped Borel Anosov representation from a lattice Γ ⊂ PGL 2 (R) to PGL d (R) contains a unipotent element with a single Jordan block in its image, then it is necessarily a (cusped) Hitchin representation. We also show that the amalgamation of a Hitchin representation with a cusped Borel Anosov representation that is not Hitchin is never cusped Borel Anosov. [ABSTRACT FROM AUTHOR]

We survey the theory of Hitchin representations of closed surface groups into PSL (d,R) with a focus on their dynamical and geometric properties. We then describe recent extensions of this work to study Hitchin representations of co-finite area Fuchsian groups. The motivation for this recent work is a conjecture about the geometry of the augmented Hitchin component. [ABSTRACT FROM AUTHOR]

Current research builds labelings for geometrically uniform codes on the double torus through tiling groups. At least one labeling group was provided for all of the 11 regular tessellations on the double torus, derived from triangular Fuchsian groups, as well as extensions of these labeling groups to generate new codes. An important consequence is that such techniques can be used to label geometrically uniform codes on surfaces with greater genera. Furthermore, partitioning chains are constructed into geometrically uniform codes using soluble groups as labeling, which in some cases results in an Ungerboeck partitioning for the surface. As a result of these constructions, it is demonstrated that, as in Euclidean spaces, modulation and encoding can be combined in a single step in hyperbolic space. [ABSTRACT FROM AUTHOR]

In this article, we extend results of Zomorrodian to determine upper bounds for the order of a nilpotent group of automorphisms of a complex d-dimensional family of compact Riemann surfaces, where d ⩾ 1. We provide conditions under which these bounds are sharp. In addition, for the one-dimensional case, we construct and describe an explicit family attaining the bound for infinitely many genera. We obtain similar results for the case of p-groups of automorphisms. [ABSTRACT FROM AUTHOR]

Given a closed, orientable, compact surface S of constant negative curvature and genus $g \geq 2$ , we study the measure-theoretic entropy of the Bowen–Series boundary map with respect to its smooth invariant measure. We obtain an explicit formula for the entropy that only depends on the perimeter of the $(8g-4)$ -sided fundamental polygon of the surface S and its genus. Using this, we analyze how the entropy changes in the Teichmüller space of S and prove the following flexibility result: the measure-theoretic entropy takes all values between 0 and a maximum that is achieved on the surface that admits a regular $(8g-4)$ -sided fundamental polygon. We also compare the measure-theoretic entropy to the topological entropy of these maps and show that the smooth invariant measure is not a measure of maximal entropy. [ABSTRACT FROM AUTHOR]

Let Δ be a hyperbolic triangle with a fixed area φ . We prove that for all but countably many φ , generic choices of Δ have the property that the group generated by the π -rotations about the midpoints of the sides of the triangle admits no nontrivial relations. By contrast, we show for all φ ∈ (0 , π) \ Q π , a dense set of triangles does afford nontrivial relations, which in the generic case map to hyperbolic translations. To establish this fact, we study the deformation space C θ of singular hyperbolic metrics on a torus with a single cone point of angle θ = 2 (π - φ) , and answer an analogous question for the holonomy map ρ ξ of such a hyperbolic structure ξ . In an appendix by Gao, concrete examples of θ and ξ ∈ C θ are given where the image of each ρ ξ is finitely presented, non-free and torsion-free; in fact, those images will be isomorphic to the fundamental groups of closed hyperbolic 3-manifolds. [ABSTRACT FROM AUTHOR]

The category of smooth, irreducible, projective, complex algebraic curves is equivalent to the category of compact Riemann surfaces. We study automorphism groups of Riemann surfaces which are equivalent to complex algebraic curves with real moduli. A complex algebraic curve C has real moduli when the corresponding surface X C admits an anti-conformal automorphism. If no such an automorphism is an involution (symmetry), then the surface X C is called pseudo-real and the curve C is isomorphic to its conjugate, but is not definable over reals. Otherwise, the surface X C is called symmetric and the curve C is real. [ABSTRACT FROM AUTHOR]