Convex fundamental domains of Fuchsian groups

In this paper a Fuchsian group G shall be a discrete group of Möbius transformations each of which maps the unit disc △ in the complex plane onto itself. We shall also assume throughout this paper that G is both finitely generated and of the first kind.

Download Full-text

Limit points for infinitely generated Fuchsian groups

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100065737 ◽

1988 ◽

Vol 104 (3) ◽

pp. 539-545

Author(s):

Shunsuke Morosawa

Keyword(s):

Complex Plane ◽

Unit Disc ◽

Fuchsian Group ◽

Discrete Subgroup ◽

Fuchsian Groups ◽

Limit Set ◽

Möbius Transformations ◽

Disjoint Sets ◽

Limit Points ◽

Image Position

Let D be the unit disc in the complex plane ℂ with centre 0 and let ∂D be its boundary. By Möb (D) we denote the group of all Möbius transformations which leave D invariant. A Fuchsian group G acting on D is a discrete subgroup of Möb (D). The limit set of G is in ∂D. We decompose ∂D into the following three disjoint sets:

Download Full-text

Transitivity Properties of Fuchsian Groups

Canadian Journal of Mathematics ◽

10.4153/cjm-1976-077-8 ◽

1976 ◽

Vol 28 (4) ◽

pp. 805-814 ◽

Cited By ~ 7

Author(s):

Peter J. Nicholls

Keyword(s):

Half Plane ◽

Unit Disc ◽

Fuchsian Group ◽

Discrete Group ◽

Fuchsian Groups ◽

Linear Transforms

A Fuchsian group G is a discrete group of fractional linear transforms each of which preserve a disc (or half plane). We consider only groups which preserve the unit disc Δ = {z: |z| < 1} and none of whose transforms, except the identity, fix infinity (any Fuchsian group is conjugate to such a group).

Download Full-text

On the Images of Ellipses Under Similarities

Analele Universitatii Ovidius Constanta - Seria Matematica ◽

10.2478/auom-2013-0032 ◽

2013 ◽

Vol 21 (2) ◽

pp. 189-194

Author(s):

Nihal Yilmaz Özgür

Keyword(s):

Complex Plane ◽

Möbius Transformations ◽

Mobius Transformations

AbstractWe consider ellipses corresponding to any norm function on the complex plane and determine their images under the similarities which are special Möbius transformations.

Download Full-text

Hyperbolic Motions

Nagoya Mathematical Journal ◽

10.1017/s0027763000024259 ◽

1967 ◽

Vol 29 ◽

pp. 163-166 ◽

Cited By ~ 9

Author(s):

Lars V. Ahlfors

Keyword(s):

Half Space ◽

Hyperbolic Space ◽

Complex Plane ◽

Two Dimensions ◽

Möbius Transformations ◽

Do So ◽

Mobius Transformations

The observation by Poincaré that Möbius transformations in the complex plane can be lifted to a half-space raises the need to be able to handle motions in hyperbolic space of more than two dimensions by means of an analytic apparatus of not too forbidding complexity. In my experience the best way to do so is to be guided by analogies with the familiar twodimensional case. The purpose of this little paper is to collect a few formulas that the writer has found useful when working with certain hyperbolically invariant operators.

Download Full-text

Exceptional points for finitely generated Fuchsian groups of the first kind

Advances in Geometry ◽

10.1515/advgeom-2019-0013 ◽

2020 ◽

Vol 20 (4) ◽

pp. 523-526

Author(s):

Joseph Fera ◽

Andrew Lazowski

Keyword(s):

Fuchsian Group ◽

Fuchsian Groups ◽

Finitely Generated ◽

Exceptional Points

AbstractLet G be a finitely generated Fuchsian group of the first kind and let (g : m1, m2, …, mn) be its shortened signature. Beardon showed that almost every Dirichlet region for G has 12g + 4n − 6 sides. Points in ℍ corresponding to Dirichlet regions for G with fewer sides are called exceptional for G. We generalize previously established methods to show that, for any such G, its set of exceptional points is uncountable.

Download Full-text

Hermite Interpolation Using Möbius Transformations of Planar Pythagorean-Hodograph Cubics

Abstract and Applied Analysis ◽

10.1155/2012/560246 ◽

2012 ◽

Vol 2012 ◽

pp. 1-15 ◽

Cited By ~ 3

Author(s):

Sunhong Lee ◽

Hyun Chol Lee ◽

Mi Ran Lee ◽

Seungpil Jeong ◽

Gwang-Il Kim

Keyword(s):

Complex Plane ◽

Hermite Interpolation ◽

Möbius Transformation ◽

Möbius Transformations ◽

Interpolation Problems ◽

Pythagorean Hodograph ◽

Improved Stability ◽

Mobius Transformation ◽

Extra Parameter ◽

Mobius Transformations

We present an algorithm forC1Hermite interpolation using Möbius transformations of planar polynomial Pythagoreanhodograph (PH) cubics. In general, with PH cubics, we cannot solveC1Hermite interpolation problems, since their lack of parameters makes the problems overdetermined. In this paper, we show that, for each Möbius transformation, we can introduce anextra parameterdetermined by the transformation, with which we can reduce them to the problems determining PH cubics in the complex planeℂ. Möbius transformations preserve the PH property of PH curves and are biholomorphic. Thus the interpolants obtained by this algorithm are also PH and preserve the topology of PH cubics. We present a condition to be met by a Hermite dataset, in order for the corresponding interpolant to be simple or to be a loop. We demonstrate the improved stability of these new interpolants compared with PH quintics.

Download Full-text

Convex fundamental regions for Fuchsian groups: II

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100056127 ◽

1979 ◽

Vol 86 (2) ◽

pp. 295-300

Author(s):

P. Nicholls ◽

R. Zarrow

Keyword(s):

Unit Disc ◽

Fundamental Domain ◽

Fuchsian Groups ◽

Finitely Generated ◽

Locally Finite ◽

Open Set

1.Introduction. In this article we continue the work begun in (5). We will consider only finitely generated Fuchsian groups of the first kind. LetGbe such a group acting on the unit disc Δ. A fundamental domainDforGis a connected open set with the property that any point of Δ isG-equivalent to exactly one point inDor at least one point in(the closure ofDin Δ). A fundamental domain is said to belocally finiteif there are no points in Δ where infinitely manyG-images ofDaccumulate.

Download Full-text

Convex fundamental regions for Fuchsian groups

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100055316 ◽

1978 ◽

Vol 84 (3) ◽

pp. 507-518 ◽

Cited By ~ 1

Author(s):

P. Nicholls ◽

R. Zarrow

Keyword(s):

Unit Disc ◽

Fuchsian Group ◽

Fundamental Region ◽

Fuchsian Groups ◽

Convex Region ◽

Locally Finite

1. Introduction. Let G be a Fuchsian group which acts on the unit disc Δ. A fundamental region D for G acting in Δ is a subset of Δ such that D is open and connected and each point of Δ is G-equivalent to exactly one point in D or at least one point in D̅ (the closure of D in Δ). Throughout this paper we consider only fundamental regions which are (hyperbolically) convex. Beardon has shown (2) that it is possible for a convex fundamental region to have certain undesirable properties. It can happen that a convex region is not locally finite, i.e. there exist points of Δ where infinitely many G images of D accumulate. For a domain D we denote by F the set of such points.

Download Full-text

Sharp Lipschitz Constants for the Distance Ratio Metric

MATHEMATICA SCANDINAVICA ◽

10.7146/math.scand.a-20452 ◽

2015 ◽

Vol 116 (1) ◽

pp. 86 ◽

Cited By ~ 5

Author(s):

Slavko Simić ◽

Matti Vuorinen ◽

Gendi Wang

Keyword(s):

Euclidean Space ◽

Unit Ball ◽

Unit Disk ◽

Complex Plane ◽

Distance Ratio ◽

Möbius Transformations ◽

Lipschitz Constants ◽

The Distance Ratio Metric ◽

Mobius Transformations

We study expansion/contraction properties of some common classes of mappings of the Euclidean space $\mathsf{R}^n$, $n\ge 2$, with respect to the distance ratio metric. The first main case is the behavior of Möbius transformations of the unit ball in $\mathsf{R}^n$ onto itself. In the second main case we study the polynomials of the unit disk onto a subdomain of the complex plane. In both cases sharp Lipschitz constants are obtained.

Download Full-text

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

Complex Manifolds ◽

10.1515/coma-2020-0004 ◽

2019 ◽

Vol 7 (1) ◽

pp. 73-92

Author(s):

John A. Arredondo ◽

Camilo Ramírez Maluendas

Keyword(s):

Riemann Surface ◽

Infinite Number ◽

Fuchsian Group ◽

Quotient Space ◽

Fuchsian Groups ◽

Precise Description ◽

Möbius Transformations ◽

Infinite Set ◽

Hyperbolic Riemann Surface ◽

Hyperbolic Polygon

AbstractIn 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.

Download Full-text