On Ruelle’s property

2021 ◽  
pp. 1-13
Author(s):  
SHENGJIN HUO ◽  
MICHEL ZINSMEISTER
Keyword(s):  
Fuchsian Group ◽  
Fuchsian Groups ◽  
Finitely Generated ◽  
Teichmüller Spaces ◽  
Teichmuller Spaces ◽  
Negative Results ◽  
Convergence Type ◽  
Mostow Rigidity

Abstract In this paper we investigate the range of validity of Ruelle’s property. First, we show that every finitely generated Fuchsian group has Ruelle’s property. We also prove the existence of an infinitely generated Fuchsian group satisfying Ruelle’s property. Concerning the negative results, we first generalize Astala and Zinsmeister’s results [Mostow rigidity and Fuchsian groups. C. R. Math. Acad. Sci. Paris311 (1990), 301–306; Teichmüller spaces and BMOA. Math. Ann.289 (1991), 613–625] by proving that all convergence-type Fuchsian groups of the first kind fail to have Ruelle’s property. Finally, we give some results about second-kind Fuchsian groups.

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 On Teichmüller Spaces of Koebe Groups

1997 ◽  
Vol 08 (05) ◽  
pp. 611-632
Author(s):  
Pablo Arés Gastesi
Keyword(s):  
Teichmüller Space ◽  
Universal Covering ◽  
Kleinian Groups ◽  
Fuchsian Groups ◽  
Teichmuller Space ◽  
Isomorphism Theorem ◽  
Teichmüller Spaces ◽  
Teichmuller Spaces ◽  
Covering Group

In this paper we study the Teichmüller space of constructible Koebe groups. These are Kleinian groups arising from planar covering of 2-orbifolds. In the first part, we parametrize the Teichmüller spaces of Koebe groups using a technique that can be applied to explicitly compute generators of these groups, maybe by programming a computer. In the second part, we study some properties of these Teichmüller spaces. More precisely, we find the covering group of these spaces (the universal covering is the Teichmüller space of the punctured surface), and prove an isomorphism theorem similar to the Bers–Greenberg theorem for Fuchsian groups.

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.

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