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).

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.

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:

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.

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.

scholarly journals A pointwise ergodic theorem for Fuchsian groups

2011 ◽  
Vol 151 (1) ◽  
pp. 145-159 ◽  
Author(s):  
ALEXANDER I. BUFETOV ◽  
CAROLINE SERIES
Keyword(s):  
Ergodic Theorem ◽  
Fuchsian Group ◽  
Fuchsian Groups ◽  
Limit Sets ◽  
Pointwise Ergodic Theorem ◽  
Cesaro Averages

AbstractWe use Series' Markovian coding for words in Fuchsian groups and the Bowen-Series coding of limit sets to prove an ergodic theorem for Cesàro averages of spherical averages in a Fuchsian group.

RATIONAL EQUIVARIANT FORMS

2012 ◽  
Vol 08 (04) ◽  
pp. 963-981 ◽  
Author(s):  
ABDELKRIM EL BASRAOUI ◽  
ABDELLAH SEBBAR
Keyword(s):  
Half Plane ◽  
Modular Forms ◽  
Discrete Group ◽  
Genus Zero ◽  
Modular Subgroup ◽  
Upper Half Plane ◽  
Schwarz Derivative

We investigate the notion of equivariant forms as functions on the upper half-plane commuting with the action of a discrete group. We put an emphasis on the rational equivariant forms for a modular subgroup that are parametrized by generalized modular forms. Furthermore, we study this parametrization when the modular subgroup is of genus zero as well as their behavior under the effect of the Schwarz derivative.

scholarly journals Infinite Paths of Minimal Length on Suborbital Graphs for Some Fuchsian Groups

2019 ◽  
Vol 2019 ◽  
pp. 1-8
Author(s):  
Khuanchanok Chaichana ◽  
Pradthana Jaipong
Keyword(s):  
Prime Number ◽  
Fuchsian Group ◽  
Continued Fraction ◽  
Sufficient Conditions ◽  
Minimal Length ◽  
Necessary And Sufficient Conditions ◽  
Fuchsian Groups ◽  
Suborbital Graphs ◽  
Necessary And Sufficient ◽  
Recursive Representation

In this study, we work on the Fuchsian group Hm where m is a prime number acting on mℚ^ transitively. We give necessary and sufficient conditions for two vertices to be adjacent in suborbital graphs induced by these groups. Moreover, we investigate infinite paths of minimal length in graphs and give the recursive representation of continued fraction of such vertex.

The Values of Modular Functions and Modular Forms

2006 ◽  
Vol 49 (4) ◽  
pp. 526-535 ◽  
Author(s):  
So Young Choi
Keyword(s):  
Modular Form ◽  
Half Plane ◽  
Finite Index ◽  
Modular Forms ◽  
Fuchsian Group ◽  
Modular Functions ◽  
Genus Zero ◽  
Recursive Formulas ◽  
Upper Half Plane ◽  
Recursive Relations

AbstractLet Γ0 be a Fuchsian group of the first kind of genus zero and Γ be a subgroup of Γ0 of finite index of genus zero. We find universal recursive relations giving the qr-series coefficients of j0 by using those of the qhs -series of j, where j is the canonical Hauptmodul for Γ and j0 is a Hauptmodul for Γ0 without zeros on the complex upper half plane (here qℓ := e2πiz/ℓ). We find universal recursive formulas for q-series coefficients of any modular form on in terms of those of the canonical Hauptmodul .

Composition operators on weighted spaces of holomorphic functions on the upper half plane

2018 ◽  
Vol 122 (1) ◽  
pp. 141
Author(s):  
Wolfgang Lusky
Keyword(s):  
Half Plane ◽  
Composition Operator ◽  
Unit Disc ◽  
Composition Operators ◽  
Bounded Operator ◽  
Holomorphic Functions ◽  
Weighted Spaces ◽  
Weight Functions ◽  
Spaces Of Holomorphic Functions ◽  
Upper Half Plane

We consider moderately growing weight functions $v$ on the upper half plane $\mathbb G$ called normal weights which include the examples $(\mathrm{Im} w)^a$, $w \in \mathbb G$, for fixed $a > 0$. In contrast to the comparable, well-studied situation of normal weights on the unit disc here there are always unbounded composition operators $C_{\varphi }$ on the weighted spaces $Hv(\mathbb G)$. We characterize those holomorphic functions $\varphi \colon \mathbb G \rightarrow \mathbb G$ where the composition operator $C_{\varphi } $ is a bounded operator $Hv(\mathbb G) \rightarrow Hv(\mathbb G)$ by a simple property which depends only on $\varphi $ but not on $v$. Moreover we show that there are no compact composition operators $C_{\varphi }$ on $Hv(\mathbb G)$.

Transitivity for the modular group

1980 ◽  
Vol 88 (3) ◽  
pp. 409-423 ◽  
Author(s):  
Mark Sheingorn
Keyword(s):  
Real Number ◽  
Half Plane ◽  
Limit Point ◽  
Fuchsian Group ◽  
Modular Group ◽  
Fundamental Region ◽  
Hyperbolic Line ◽  
Upper Half Plane

Let Γ be a Fuchsian group of the first kind acting on the upper half plane H+. Let be a Ford fundamental region for Γ in H+. Let ξ be a real number (a limit point) and let L( = Lξ) = {ξ + iy|0 ≤ y < 1}. L can be broken into successive intervals each one of which can be mapped by an element of Γ into . Since L is a hyperbolic line (h-line), this gives us a set of h-arcs in which we will call the image.

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