The hyperbolic polygons of type (ϵ, n) and Möbius transformations

An n-sided hyperbolic polygon of type (ϵ, n) is a hyperbolic polygon with ordered interior angles $\begin{array}{} \frac{\pi}{2} \end{array} $ + ϵ, θ1, θ2, …, θn−2, $\begin{array}{} \frac{\pi}{2} \end{array} $ − ϵ, where 0 < ϵ < $\begin{array}{} \frac{\pi}{2} \end{array} $ and 0 < θi < π satisfying $$\begin{array}{} \displaystyle \sum_{i = 1}^{n-2} \theta_{i}+\Big(\frac{\pi}{2}+\epsilon\Big)+\Big(\frac{\pi}{2}-\epsilon\Big) \lt (n-2)\pi \end{array} $$ and θi + θi+1 ≠ π (1 ≤ i ≤ n − 3), θ1 + ( $\begin{array}{} \frac{\pi}{2} \end{array} $ + ϵ) ≠ π, θn−2 + ( $\begin{array}{} \frac{\pi}{2} \end{array} $ − ϵ) ≠ π. In this paper, we present a new characterization of Möbius transformations by using n-sided hyperbolic polygons of type (ϵ, n). Our proofs are based on a geometric approach.

On Infinitely generated Fuchsian groups of the Loch Ness monster, the Cantor tree and the Blooming Cantor tree

In this paper, for a non-compact Riemman surface S homeomorphic to either: the Infinite Loch Ness monster, the Cantor tree and the Blooming Cantor tree, we give a precise description of an infinite set of generators of a Fuchsian group Γ < PSL(2, ℝ), such that the quotient space ℍ/Γ is a hyperbolic Riemann surface homeomorphic to S. For each one of these constructions, we exhibit a hyperbolic polygon with an infinite number of sides and give a collection of Mobius transformations identifying the sides in pairs.

EFFICIENT GENERATION OF HYPERBOLIC PATTERNS FROM A SINGLE ASYMMETRIC MOTIF

We present an efficient method of constructing hyperbolic patterns based on an asymmetric motif designed in the central hyperbolic polygon. Since there is no rotational symmetry in each hyperbolic polygon, a subset of the hyperbolic group elements has to be selected carefully so that the central hyperbolic polygon is transformed to the other polygons once and only once. An efficient labeling procedure is proved by considering the group presentation and can be easily implemented using the computer. Illustrative hyperbolic patterns are constructed from given asymmetric motifs for the symmetry group [[Formula: see text], [Formula: see text]][Formula: see text] which consists of all compositions of an even number of reflections.

Kazhdan–Lusztig cells in planar hyperbolic Coxeter groups and automata

Let C be a one- or two-sided Kazhdan–Lusztig cell in a Coxeter group (W, S), and let Red (C) be the set of reduced expressions of all w ∈ C, regarded as a language over the alphabet S. Casselman has conjectured that Red (C) is regular. In this paper, we give a conjectural description of the cells when W is the group corresponding to a hyperbolic polygon, and show that our conjectures imply Casselman's.

Codes over Graphs Derived from Quotient Rings of the Quaternion Orders

We propose the construction of signal space codes over the quaternion orders from a graph associated with the arithmetic Fuchsian group Γ8. This Fuchsian group consists of the edge-pairing isometries of the regular hyperbolic polygon (fundamental region) P8, which tessellates the hyperbolic plane 𝔻2. Knowing the generators of the quaternion orders which realize the edge pairings of the polygon, the signal points of the signal constellation (geometrically uniform code) derived from the graph associated with the quotient ring of the quaternion order are determined.

CONVEX STANDARD FUNDAMENTAL DOMAIN FOR SUBGROUPS OF HECKE GROUPS

It is well known that if a convex hyperbolic polygon is constructed as a fundamental domain for a subgroup of SL(2,ℝ), then its translates by the group form a locally finite tessellation and its side-pairing transformations form a system of generators for the group. Such a hyperbolically convex fundamental domain for any discrete subgroup can be obtained by using Dirichlet’s and Ford’s polygon constructions. However, these two results are not well adapted for the actual construction of a hyperbolically convex fundamental domain due to their nature of construction. A third, and most important and practical, method of obtaining a fundamental domain is through the use of a right coset decomposition as described below. If Γ2 is a subgroup of Γ1 such that Γ1=Γ2⋅{L1,L2,…,Lm} and 𝔽 is the closure of a fundamental domain of the bigger group Γ1, then the set is a fundamental domain of Γ2. One can ask at this juncture, is it possible to choose the right coset suitably so that the set ℛ is a convex hyperbolic polygon? We will answer this question affirmatively for Hecke modular groups.

UNICELLULAR DESSINS AND A UNIQUENESS THEOREM FOR KLEIN'S RIEMANN SURFACE OF GENUS 3

If we consider the 14-sided hyperbolic polygon of Felix Klein that defines his famous surface of genus 3, then we observe that we have a uniform, unifacial dessin whose automorphism group is transitive on the edges, but not on the directed edges of the dessin. We show that Klein's surface is the unique platonic surface with this property.