Finitely generated fuchsian groups.

1987 ◽  
Vol 1987 (375-376) ◽  
pp. 394-405
Keyword(s):  
Fuchsian Groups ◽  
Finitely Generated

On finitely generated fuchsian groups

1967 ◽  
Vol 42 (1) ◽  
pp. 81-85 ◽  
Author(s):  
Albert Marden
Keyword(s):  
Fuchsian Groups ◽  
Finitely Generated

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 Divergence type of some subgroups of finitely generated Fuchsian groups

1981 ◽  
Vol 1 (2) ◽  
pp. 209-221 ◽  
Author(s):  
Mary Rees
Keyword(s):  
Critical Exponent ◽  
Hyperbolic Plane ◽  
Sufficient Conditions ◽  
Discrete Subgroup ◽  
Necessary And Sufficient Conditions ◽  
Fuchsian Groups ◽  
Finitely Generated ◽  
Parabolic Element ◽  
Divergence Type ◽  
Necessary And Sufficient

AbstractLet Г be a finitely generated discrete subgroup of the isometries of the hyperbolic plane H2 with at least one parabolic element. We prove that, if Г1 is a subgroup of Г with Г/Г1 abelian, the ‘critical exponent’ of Г1 is the same as that of Г. We give necessary and sufficient conditions-in terms of the rank of Г/Г1, the critical exponent of Г, and the image of parabolic elements of Г in Г/Г1 - for the Poincaré series of Г1 to diverge at the critical exponent.

Exceptional points for finitely generated Fuchsian groups of the first kind

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.

Convex fundamental regions for Fuchsian groups: II

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.

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