The Geometry of Discrete Groups

Beardon, Alan F.

doi:10.1007/978-1-4612-1146-4

Cited by 1,274 publications

(1,774 citation statements)

References 0 publications

Supporting

Mentioning

1,692

Contrasting

Unclassified

Order By: Relevance

“…Another well-known result in the same direction is Jørgensen's inequality [12,134]: If f, g are elements in PSL(2, R) generating a Fuchsian group of second kind, then…”

Section: A Sharp C 1 Version Via Lyapunov Exponentsmentioning

confidence: 99%

Groups of Circle Diffeomorphisms

Navas¹

2011

148

171

View full text Add to dashboard Cite

The original version (in Spanish) of this text was prepared for a mini-course in Antofagasta, Chile. Subsequently, enlarged and revised versions were published in the series Monografías del IMCA (Perú) and Ensaios Matemáticos (Brasil). This translation arose from the necessity of making this text accessible to a larger audience. I would like to thank Juan Rivera-Letelier for his invitation to the II Workshop on Dynamical Systems ( 2001), for which the original version was prepared, and Roger Metzger for his invitation to IMCA (2006), where part of this material was presented. I would also like to thank both Étienne Ghys and Maria Eulália Vares for motivating me and allowing me to publish the revised Spanish version in Brasil, as well as all the people who strongly encouraged me to conclude this English version.This work was partially supported by CONICYT and PBCT via the Research Network on Low Dimensional Dynamics. I would also to acknowledge the support of both the UMPA Department of the École Normale Supérieure de Lyon, where the idea of writing this text was born during my PhD thesis, and the Institut des Hautes Études Scientifiques, where the original notes started taking its definite form while I benefited from a one-year postdoctoral position.This work owes a lot to many of my colleges. Several remarks spread throughout the text and the content of some examples and exercises were born during fruitful and quite stimulating discussions. It is then a pleasure to thank Sylvain Crovisier (Proposition 4.2.25),

show abstract

“…Another well-known result in the same direction is Jørgensen's inequality [12,134]: If f, g are elements in PSL(2, R) generating a Fuchsian group of second kind, then…”

Section: A Sharp C 1 Version Via Lyapunov Exponentsmentioning

confidence: 99%

Groups of Circle Diffeomorphisms

Navas¹

2011

148

171

View full text Add to dashboard Cite

show abstract

“…Let μ = |γ |, and let us consider the ray joining the origin of the axes to the point of intersection between h and h 1 , determining an angle μ between the axis of h 3 and this radius. Then, by using hyperbolic geometry, μ is defined by coth μ = cosh μ (see [3]). …”

Section: Construction Of Fundamental Domain and Definition Of Boundarmentioning

confidence: 99%

Systoles on Compact Riemann Surfaces with Symbolic Dynamics

Grácio

2015

Springer Proceedings in Mathematics &Amp; Statistics

View full text Add to dashboard Cite

In this chapter, systolic inequalities are established, precise values are computed, and their behavior is also examined with the variation of the FenchelNielsen coordinates on a compact Riemann surface of genus 2. IntroductionThe metric and geometric structure of surfaces may be studied by using the closed geodesics spectrum and the Laplace-Beltrami operator spectrum. It is not easy to obtain these spectra and even more difficult is to describe their dependence on the parameters which determine the metric and geometric structure of a surface. The dependence of such spectra dependence is examined using a boundary map when a Riemann surface M of genus 2, thus with negative curvature, is considered.From a classical point of view the hyperelliptic surfaces are the simplest Riemann surfaces [12]. They can be denned by an algebraic curve y 2 = F (x) where F (x) is a polynomial of degree 2τ + 1 or 2τ + 2 with distinct roots (τ is the genus of the surface). Hyperelliptic surfaces of genus τ are characterized by the fact that the number of different Weierstrass points is minimal, namely 2τ + 2 (the fixed points of the hyperelliptic involution), while the weight of each Weierstrass point is maximal, namely 1 2 τ (τ − 1). For our purposes, two results for surfaces (see [17] The systole of a compact Riemann surface is defined as the minimum length of a noncontractile curve. In the 1990s, a number of studies developed this concept: in particular, the article published by Schmutz Schaller (see [17]) that spurred the C. Grácio ( ) CIMA-UE-DMAT,

show abstract

“…To verify it, first suppose that X is a loxodromic element in Γ whose axis projects to a closed geodesic on H 3 /Γ of length L ≥ 3 (see, for instance, Section 1.3 of [22]). Since |trX| = 2 cosh L 2 (Section 7.34 of [3]) and hyperbolic cosine is an increasing function, L ≥ 3 implies |trX| ≥ 2 cosh 3 2 = 4.704819230.…”

Section: Computations Of Jørgensen Numbersmentioning

confidence: 99%

Jørgensen number and arithmeticity

Callahan

2009

Conform. Geom. Dyn.

View full text Add to dashboard Cite

Abstract. The Jørgensen number of a rank-two non-elementary Kleinian group Γ isJørgensen's Inequality guarantees J(Γ) ≥ 1, and Γ is a Jørgensen group if J(Γ) = 1. This paper shows that the only torsion-free Jørgensen group is the figure-eight knot group, identifies all non-cocompact arithmetic Jørgensen groups, and establishes a characterization of cocompact arithmetic Jørgensen groups. The paper concludes with computations of J(Γ) for several noncocompact Kleinian groups including some two-bridge knot and link groups.

show abstract

The Geometry of Discrete Groups

Cited by 1,274 publications

References 0 publications

Groups of Circle Diffeomorphisms

Groups of Circle Diffeomorphisms

Systoles on Compact Riemann Surfaces with Symbolic Dynamics

Jørgensen number and arithmeticity

Contact Info

Product

Resources

About