Limit points for infinitely generated Fuchsian groups

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:

Convex fundamental domains of Fuchsian groups

1974 ◽  
Vol 76 (3) ◽  
pp. 511-513 ◽  
Author(s):  
A. F. Beardon
Keyword(s):  
Complex Plane ◽  
Unit Disc ◽  
Fuchsian Group ◽  
Discrete Group ◽  
Fuchsian Groups ◽  
Finitely Generated ◽  
Möbius Transformations ◽  
Fundamental Domains ◽  
Mobius Transformations

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.

scholarly journals Certain nth Order Differential Inequalities in the Complex Plane

1978 ◽  
Vol 21 (3) ◽  
pp. 273-277
Author(s):  
H. S. Al-Amiri
Keyword(s):  
Differential Equations ◽  
Complex Plane ◽  
Unit Disc ◽  
Hadamard Product ◽  
Complex Function ◽  
Differential Inequalities ◽  
Image Position

AbstractLet w(z) be regular in the unit disc U:|z|<l, with w(0) = 0 and let h(r, s, t) be a complex function defined in a domain D of C3. The author determines conditions on h such that ifz∈U, then |w(z)|< 1 for z ∈ U and n= 0, 1, 2, …. Here Dnw(z) = (z/(l-z)n+1*w(z), where * stands for the Hadamard product (convolution). Some applications of the results to certain differential equations are given.

Transitivity Properties of Fuchsian Groups

1976 ◽  
Vol 28 (4) ◽  
pp. 805-814 ◽  
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).

A modular version of Jensen's formula

1984 ◽  
Vol 95 (1) ◽  
pp. 15-20 ◽  
Author(s):  
David E. Rohrlich
Keyword(s):  
Meromorphic Function ◽  
Modular Forms ◽  
Unit Disc ◽  
Function Theory ◽  
Modular Function ◽  
Fuchsian Groups ◽  
Double Integral ◽  
Kronecker Limit Formula ◽  
Classical Function ◽  
Image Position

A well-known result of classical function theory, Jensen's formula, expresses the integral around a circle of the log modulus of a meromorphic function in terms of the log modulus of the zeros and poles of that function lying inside the circle. Explicitly, if F is a meromorphic function on the unit disc {ω ε ℂ: |ω| < 1} and F(0) = 1, then, for 0 < r < 1,where ordωF is the order of F at ω. The purpose of this note is to observe that a formula analogous to (1) holds when F is replaced by a modular function for SL2(ℤ) and the integral by a suitable double integral over a fundamental domain. We shall derive this modular variant of Jensen's formula from the usual version by applying the Rankin-Selberg method and the first Kronecker limit formula. The argument admits some extension to Fuchsian groups other than SL2(ℤ), and to modular forms of weight other than zero; this point will be discussed later.

scholarly journals The Closed Ideals in an Algebra of Analytic Functions

1957 ◽  
Vol 9 ◽  
pp. 426-434 ◽  
Author(s):  
Walter Rudin
Keyword(s):  
Banach Algebra ◽  
Complex Plane ◽  
Analytic Functions ◽  
Unit Disc ◽  
Open Unit ◽  
Open Unit Disc ◽  
Complex Functions ◽  
Continuous Complex ◽  
Image Position

Let K and C be the closure and boundary, respectively, of the open unit disc U in the complex plane. Let be the Banach algebra whose elements are those continuous complex functions on K which are analytic in U, with norm (f ∊ ).

Ford and Dirichlet Regions for Fuchsian Groups

1982 ◽  
Vol 34 (4) ◽  
pp. 806-815 ◽  
Author(s):  
A. F. Beardon ◽  
P. J. Nicholls
Keyword(s):  
Half Plane ◽  
Unit Disc ◽  
Hyperbolic Plane ◽  
Fuchsian Groups ◽  
Linear Measure ◽  
Open Disc ◽  
Upper Half Plane ◽  
The Subject ◽  
Extended Complex ◽  
Limit Points

There has recently been some interest in a class of limit points for Fuchsian groups now known as Garnett points [5], [8]. In this paper we show that such points are intimately connected with the structure of Dirichlet regions and the same ideas serve to show that the Ford and Dirichlet regions are merely examples of one single construction which also yields fundamental regions based at limit points (and which properly lies in the subject of inversive geometry). We examine in the general case how the region varies continuously with the construction. Finally, we consider the linear measure of the set of Garnett points.2. Hyperbolic space.Let Δ be any open disc (or half-plane) in the extended complex planeC∞: usually Δ will be the unit disc or the upper half-plane. We may regard Δ as the hyperbolic plane in the usual way and the conformai isometries of Δ are simply the Moebius transformations of Δ onto itself.

Riemann Surfaces as Orbit Spaces of Fuchsian Groups

1972 ◽  
Vol 24 (4) ◽  
pp. 612-616 ◽  
Author(s):  
M. J. Moore
Keyword(s):  
Riemann Surface ◽  
Fuchsian Group ◽  
Riemann Surfaces ◽  
Hyperbolic Group ◽  
Discrete Subgroup ◽  
Orbit Space ◽  
Analytic Structure ◽  
Linear Fractional Transformations ◽  
Upper Half Plane ◽  
Image Position

A Fuchsian group is a discrete subgroup of the hyperbolic group, L.F. (2, R), of linear fractional transformationseach such transformation mapping the complex upper half plane D into itself. If Γ is a Fuchsian group, the orbit space D/Γ has an analytic structure such that the projection map p: D → D/Γ, given by p(z) = Γz, is holomorphic and D/Γ is then a Riemann surface.If N is a normal subgroup of a Fuchsian group Γ, then N is a Fuchsian group and S = D/N is a Riemann surface. The factor group, G = Γ/N, acts as a group of automorphisms (biholomorphic self-transformations) of S for, if γ ∊ Γ and z ∊ D, then γN ∊ G, Nz ∊ S, and (γN) (Nz) = Nγz. This is easily seen to be independent of the choice of γ in its N-coset and the choice of z in its N-orbit.Conversely, if S is a compact Riemann surface, of genus at least two, then S can be identified with D/K, where K is a Fuchsian group acting without fixed points in D.

Evaluation operators on the Bergman space

1995 ◽  
Vol 117 (3) ◽  
pp. 513-523 ◽  
Author(s):  
Kehe Zhu
Keyword(s):  
Bergman Space ◽  
Complex Plane ◽  
Analytic Functions ◽  
Unit Disc ◽  
Open Unit ◽  
Open Unit Disc ◽  
Space Of Analytic Functions ◽  
Area Measure ◽  
Image Position

Let D be the open unit disc in the complex plane C and let dA be the normalized area measure on D. The Bergman space is the space of analytic functions f in D such that

Convex fundamental regions for Fuchsian groups

1978 ◽  
Vol 84 (3) ◽  
pp. 507-518 ◽  
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.

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