Kleinian Groups in Higher Dimensions
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Möbius Transformations
Möbius transformations Möbius transformations are simply the degree one rational maps of C: az + b σ : z 7! : ! A cz + d C C where ad − bc 6= 0 and a b A = c d Then A 7! σA : GL(2C) ! fMobius transformations g is a homomorphism whose kernel is ∗ fλI : λ 2 C g: The homomorphism is an isomorphism restricted to SL(2; C), the subgroup of matri- ces of determinant 1. We have an action of GL(2; C) on C by A:(B:z) = (AB):z; A; B 2 GL(2; C); z 2 C: The action of SL(2; R) preserves the upper half-plane fz 2 C : Im(z) > 0g and also R [ f1g and the lower half-plane. The action of the subgroup a b SU(1; 1) = : jaj2 − jbj2 = 1 b a preserves the open unit disc, the closed unit disc, and its exterior. All of these actions are transitive that is, for all z and w in the domain there is A in the group with A:z = w. Kleinian groups Definition 1 A Kleinian group is a subgroup Γ of P SL(2; C) which is discrete, that is, there is an open neighbourhood U ⊂ P SL(2; C) of the identity element I such that U \ Γ = fIg: Definition 2 A Fuchsian group is a discrete subgroup of P SL(2; R) Equivalently (as usual with topological groups) there is an open neighbourhood V of I such that γV \ γ0V = ; for all γ, γ0 2 Γ, γ 6= γ0. To get this, choose V with V = V −1 and V:V ⊂ U. -
A Brief Survey of the Deformation Theory of Kleinian Groups 1
ISSN 1464-8997 (on line) 1464-8989 (printed) 23 Geometry & Topology Monographs Volume 1: The Epstein birthday schrift Pages 23–49 A brief survey of the deformation theory of Kleinian groups James W Anderson Abstract We give a brief overview of the current state of the study of the deformation theory of Kleinian groups. The topics covered include the definition of the deformation space of a Kleinian group and of several im- portant subspaces; a discussion of the parametrization by topological data of the components of the closure of the deformation space; the relationship between algebraic and geometric limits of sequences of Kleinian groups; and the behavior of several geometrically and analytically interesting functions on the deformation space. AMS Classification 30F40; 57M50 Keywords Kleinian group, deformation space, hyperbolic manifold, alge- braic limits, geometric limits, strong limits Dedicated to David Epstein on the occasion of his 60th birthday 1 Introduction Kleinian groups, which are the discrete groups of orientation preserving isome- tries of hyperbolic space, have been studied for a number of years, and have been of particular interest since the work of Thurston in the late 1970s on the geometrization of compact 3–manifolds. A Kleinian group can be viewed either as an isolated, single group, or as one of a member of a family or continuum of groups. In this note, we concentrate our attention on the latter scenario, which is the deformation theory of the title, and attempt to give a description of various of the more common families of Kleinian groups which are considered when doing deformation theory. -
Handlebody Orbifolds and Schottky Uniformizations of Hyperbolic 2-Orbifolds
proceedings of the american mathematical society Volume 123, Number 12, December 1995 HANDLEBODY ORBIFOLDS AND SCHOTTKY UNIFORMIZATIONS OF HYPERBOLIC 2-ORBIFOLDS MARCO RENI AND BRUNO ZIMMERMANN (Communicated by Ronald J. Stern) Abstract. The retrosection theorem says that any hyperbolic or Riemann sur- face can be uniformized by a Schottky group. We generalize this theorem to the case of hyperbolic 2-orbifolds by giving necessary and sufficient conditions for a hyperbolic 2-orbifold, in terms of its signature, to admit a uniformization by a Kleinian group which is a finite extension of a Schottky group. Equiva- lent^, the conditions characterize those hyperbolic 2-orbifolds which occur as the boundary of a handlebody orbifold, that is, the quotient of a handlebody by a finite group action. 1. Introduction Let F be a Fuchsian group, that is a discrete group of Möbius transforma- tions—or equivalently, hyperbolic isometries—acting on the upper half space or the interior of the unit disk H2 (Poincaré model of the hyperbolic plane). We shall always assume that F has compact quotient tf = H2/F which is a hyperbolic 2-orbifold with signature (g;nx,...,nr); here g denotes the genus of the quotient which is a closed orientable surface, and nx, ... , nr are the orders of the branch points (the singular set of the orbifold) of the branched covering H2 -» H2/F. Then also F has signature (g;nx,... , nr), and (2 - 2g - £-=1(l - 1/«/)) < 0 (see [12] resp. [15] for information about orbifolds resp. Fuchsian groups). In case r — 0, or equivalently, if the Fuch- sian group F is without torsion, the quotient H2/F is a hyperbolic or Riemann surface without branch points. -
Hyperbolic Geometry, Fuchsian Groups, and Tiling Spaces
HYPERBOLIC GEOMETRY, FUCHSIAN GROUPS, AND TILING SPACES C. OLIVARES Abstract. Expository paper on hyperbolic geometry and Fuchsian groups. Exposition is based on utilizing the group action of hyperbolic isometries to discover facts about the space across models. Fuchsian groups are character- ized, and applied to construct tilings of hyperbolic space. Contents 1. Introduction1 2. The Origin of Hyperbolic Geometry2 3. The Poincar´eDisk Model of 2D Hyperbolic Space3 3.1. Length in the Poincar´eDisk4 3.2. Groups of Isometries in Hyperbolic Space5 3.3. Hyperbolic Geodesics6 4. The Half-Plane Model of Hyperbolic Space 10 5. The Classification of Hyperbolic Isometries 14 5.1. The Three Classes of Isometries 15 5.2. Hyperbolic Elements 17 5.3. Parabolic Elements 18 5.4. Elliptic Elements 18 6. Fuchsian Groups 19 6.1. Defining Fuchsian Groups 20 6.2. Fuchsian Group Action 20 Acknowledgements 26 References 26 1. Introduction One of the goals of geometry is to characterize what a space \looks like." This task is difficult because a space is just a non-empty set endowed with an axiomatic structure|a list of rules for its points to follow. But, a priori, there is nothing that a list of axioms tells us about the curvature of a space, or what a circle, triangle, or other classical shapes would look like in a space. Moreover, the ways geometry manifests itself vary wildly according to changes in the axioms. As an example, consider the following: There are two spaces, with two different rules. In one space, planets are round, and in the other, planets are flat. -
Computing Fundamental Domains for Arithmetic Kleinian Groups
Computing fundamental domains for arithmetic Kleinian groups Aurel Page 2010 A thesis submitted in partial fulfilment of the requirements of the degree of Master 2 Math´ematiques Fondamentales Universit´eParis 7 - Diderot Committee: John Voight Marc Hindry A fundamental domain for PSL Z √ 5 2 − The pictures contained in this thesis were realized with Maple code exported in EPS format, or pstricks code produced by LaTeXDraw. Acknowledgements I am very grateful to my thesis advisor John Voight, for introducing the wonder- ful world of quaternion algebras to me, providing me a very exciting problem, patiently answering my countless questions and guiding me during this work. I would like to thank Henri Darmon for welcoming me in Montreal, at the McGill University and at the CRM during the special semester; it was a very rewarding experience. I would also like to thank Samuel Baumard for his careful reading of the early versions of this thesis. Finally, I would like to give my sincerest gratitude to N´eph´eli, my family and my friends for their love and support. Contents I Kleinian groups and arithmetic 1 1 HyperbolicgeometryandKleiniangroups 1 1.1 Theupperhalf-space......................... 1 1.2 ThePoincar´eextension ....................... 2 1.3 Classification of elements . 3 1.4 Kleiniangroups............................ 4 1.5 Fundamentaldomains ........................ 5 2 QuaternionalgebrasandKleiniangroups 9 2.1 Quaternionalgebras ......................... 9 2.2 Splitting................................ 11 2.3 Orders................................. 12 2.4 Arithmetic Kleinian groups and covolumes . 15 II Algorithms for Kleinian groups 16 3 Algorithmsforhyperbolicgeometry 16 3.1 The unit ball model and explicit formulas . 16 3.2 Geometriccomputations . 22 3.3 Computinganexteriordomain . -
THREE DIMENSIONAL MANIFOLDS, KLEINIAN GROUPS and HYPERBOLIC GEOMETRY 1. a Conjectural Picture of 3-Manifolds. a Major Thrust Of
BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY Volume 6, Number 3, May 1982 THREE DIMENSIONAL MANIFOLDS, KLEINIAN GROUPS AND HYPERBOLIC GEOMETRY BY WILLIAM P. THURSTON 1. A conjectural picture of 3-manifolds. A major thrust of mathematics in the late 19th century, in which Poincaré had a large role, was the uniformization theory for Riemann surfaces: that every conformai structure on a closed oriented surface is represented by a Riemannian metric of constant curvature. For the typical case of negative Euler characteristic (genus greater than 1) such a metric gives a hyperbolic structure: any small neighborhood in the surface is isometric to a neighborhood in the hyperbolic plane, and the surface itself is the quotient of the hyperbolic plane by a discrete group of motions. The exceptional cases, the sphere and the torus, have spherical and Euclidean structures. Three-manifolds are greatly more complicated than surfaces, and I think it is fair to say that until recently there was little reason to expect any analogous theory for manifolds of dimension 3 (or more)—except perhaps for the fact that so many 3-manifolds are beautiful. The situation has changed, so that I feel fairly confident in proposing the 1.1. CONJECTURE. The interior of every compact ^-manifold has a canonical decomposition into pieces which have geometric structures. In §2, I will describe some theorems which support the conjecture, but first some explanation of its meaning is in order. For the purpose of conservation of words, we will henceforth discuss only oriented 3-manifolds. The general case is quite similar to the orientable case. -
Hyperbolic Geometry of Riemann Surfaces
3 Hyperbolic geometry of Riemann surfaces By Theorem 1.8.8, all hyperbolic Riemann surfaces inherit the geometry of the hyperbolic plane. How this geometry interacts with the topology of a Riemann surface is a complicated business, and beginning with Section 3.2, the material will become more demanding. Since this book is largely devoted to the study of Riemann surfaces, a careful study of this interaction is of central interest and underlies most of the remainder of the book. 3.1 Fuchsian groups We saw in Proposition 1.8.14 that torsion-free Fuchsian groups and hyper- bolic Riemann surfaces are essentially the same subject. Most such groups and most such surfaces are complicated objects: usually, a Fuchsian group is at least as complicated as a free group on two generators. However, in a few exceptional cases Fuchsian groups are not complicated, whether they have torsion or not. Such groups are called elementary;we classify them in parts 1–3 of Proposition 3.1.2. Part 4 concerns the com- plicated case – the one that really interests us. Notationin 3.1.1 If A is a subset of a group G, we denote by hAi the subgroup generated by A. Propositionin 3.1.2 (Fuchsian groups) Let be a Fuchsian group. 1. If is nite, it is a cyclic group generated by a rotation about a point by 2/n radians, for some positive integer n. 2. If is innite but consists entirely of elliptic and parabolic ele- ments, then it is innite cyclic, is generated by a single parabolic element, and contains no elliptic elements. -
The Geometry and Arithmetic of Kleinian Groups
The Geometry and Arithmetic of Kleinian Groups To the memory of Fred Gehring and Colin Maclachlan Gaven J. Martin∗ Institute for Advanced Study Massey University, Auckland New Zealand email: [email protected] Abstract. In this article we survey and describe various aspects of the geometry and arithmetic of Kleinian groups - discrete nonelementary groups of isometries of hyperbolic 3-space. In particular we make a detailed study of two-generator groups and discuss the classification of the arithmetic generalised triangle groups (and their near relatives). This work is mainly based around my collaborations over the last two decades with Fred Gehring and Colin Maclachlan, both of whom passed away in 2012. There are many others involved as well. Over the last few decades the theory of Kleinian groups has flourished because of its intimate connections with low dimensional topology and geometry. We give little of the general theory and its connections with 3-manifold theory here, but focus on two main problems: Siegel's problem of identifying the minimal covolume hyperbolic lattice and the Margulis constant problem. These are both \universal constraints" on Kleinian groups { a feature of discrete isometry groups in negative curvature and include results such as Jørgensen’s inequality, the higher dimensional version of Hurwitz's 84g − 84 theorem and a number of other things. We will see that big part of the work necessary to obtain these results is in getting concrete descriptions of various analytic spaces of two-generator Kleinian groups, somewhat akin to the Riley slice. 2000 Mathematics Subject Classification: 30F40, 30D50, 51M20, 20H10 Keywords: Kleinian group, hyperbolic geometry, discrete group. -
Arxiv:2103.09350V1 [Math.DS] 16 Mar 2021 Nti Ae Esuytegopo Iainltransformatio Birational Vue
BIRATIONAL KLEINIAN GROUPS SHENGYUAN ZHAO ABSTRACT. In this paper we initiate the study of birational Kleinian groups, i.e. groups of birational transformations of complex projective varieties acting in a free, properly dis- continuous and cocompact way on an open subset of the variety with respect to the usual topology. We obtain a classification in dimension two. CONTENTS 1. Introduction 1 2. Varietiesofnon-negativeKodairadimension 6 3. Complex projective Kleinian groups 8 4. Preliminaries on groups of birational transformations ofsurfaces 10 5. BirationalKleiniangroupsareelementary 13 6. Foliated surfaces 19 7. Invariant rational fibration I 22 8. InvariantrationalfibrationII:ellipticbase 29 9. InvariantrationalfibrationIII:rationalbase 31 10. Classification and proof 43 References 47 1. INTRODUCTION 1.1. Birational Kleinian groups. 1.1.1. Definitions. According to Klein’s 1872 Erlangen program a geometry is a space with a nice group of transformations. One of the implementations of this idea is Ehres- arXiv:2103.09350v1 [math.DS] 16 Mar 2021 mann’s notion of geometric structures ([Ehr36]) which models a space locally on a geom- etry in the sense of Klein. In his Erlangen program, Klein conceived a geometry modeled upon birational transformations: Of such a geometry of rational transformations as must result on the basis of the transformations of the first kind, only a beginning has so far been made... ([Kle72], [Kle93]8.1) In this paper we study the group of birational transformations of a variety from Klein and Ehresmann’s point of vue. Let Y be a smooth complex projective variety and U ⊂ Y a connected open set in the usual topology. Let Γ ⊂ Bir(Y ) be an infinite group of birational transformations. -
Riemann Surfaces As Orbit Spaces of Fuchsian Groups
Can. J. Math., Vol. XXIV, No. 4, 1972, pp. 612-616 RIEMANN SURFACES AS ORBIT SPACES OF FUCHSIAN GROUPS M. J. MOORE 1. Introduction. A Fuchsian group is a discrete subgroup of the hyperbolic group, L.F. (2, R), of linear fractional transformations w = i—7 (a> b> c> d real, ad — be = 1), cz + a each such transformation mapping the complex upper half plane D into itself. If T is a Fuchsian group, the orbit space D/T has an analytic structure such that the projection map p: D —» D/T, given by p(z) = Yz, is holomorphic and D/Y is then a Riemann surface. If N is a normal subgroup of a Fuchsian group T, then iV is a Fuchsian group and 5 = D/N is a Riemann surface. The factor group, G = T/N, acts as a group of automorphisms (biholomorphic self-transformations) of 5 for, if y e Y and z e D, then yN e G, Nz e S, and (yN) (Nz) = Nyz. This is easily seen to be independent of the choice of y in its iV-coset and the choice of z in its iV-orbit. Conversely, if 5 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. D is the universal covering space of 5 and K is isomorphic to the fundamental group of S. We shall call a Fuchsian group which acts without fixed points in D, a Fuchsian surface group. -
ON the COHOMOLOGY of FUCHSIAN GROUPS by S
ON THE COHOMOLOGY OF FUCHSIAN GROUPS by S. J. PATTERSON (Received 12 September, 1974; revised 15 May, 1975) 1. Introduction. The object of this paper is to redevelop the classical theory of multi- pliers of Fuchsian groups [16] and to attempt a classification. The language which appears most appropriate is that of group extensions and the cohomology of groups. This viewpoint is not entirely novel [12] but the entire theory has never been based on it before. A Fuchsian group G is a discrete subgroup of PSL(2 ; R). We shall further assume that G is finitely generated and that PSL(2 ; R)/G has finite volume. We shall begin by investi- gating the extensions of PSL(2 ; R). These induce extensions of G. On the other hand G can be given explicitly by generators and relations. So, in principle at any rate, its extensions can be classified from the point of view of group theory. There is then the basic problem of relating these two approaches. In the case that G has no elliptic elements, this identification can be carried out explicitly by using Chern characters to identify algebraically and geometrically defined objects. A partial extension to groups with elliptic elements can be made using the transfer homomor- phism of group cohomology. As a first application, we consider the question as to when G can be lifted to SL(2 ; R). We prove again a theorem of Petersson ([17]) and Bers ([1]) (itself the answer to a question of Siegel ([22])). Bers' proof was an application of the theory of moduli of Riemann surfaces; ours, being intrinsic, can be considered rather more satisfying. -
Fuchsian Groups, Coverings of Riemann Surfaces, Subgroup Growth, Random Quotients and Random Walks
Fuchsian groups, coverings of Riemann surfaces, subgroup growth, random quotients and random walks Martin W. Liebeck Aner Shalev Department of Mathematics Institute of Mathematics Imperial College Hebrew University London SW7 2BZ Jerusalem 91904 England Israel Abstract Fuchsian groups (acting as isometries of the hyperbolic plane) oc- cur naturally in geometry, combinatorial group theory, and other con- texts. We use character-theoretic and probabilistic methods to study the spaces of homomorphisms from Fuchsian groups to symmetric groups. We obtain a wide variety of applications, ranging from count- ing branched coverings of Riemann surfaces, to subgroup growth and random finite quotients of Fuchsian groups, as well as random walks on symmetric groups. In particular we show that, in some sense, almost all homomor- phisms from a Fuchsian group to alternating groups An are surjective, and this implies Higman’s conjecture that every Fuchsian group sur- jects onto all large enough alternating groups. As a very special case we obtain a random Hurwitz generation of An, namely random gen- eration by two elements of orders 2 and 3 whose product has order 7. We also establish the analogue of Higman’s conjecture for symmetric groups. We apply these results to branched coverings of Riemann sur- faces, showing that under some assumptions on the ramification types, their monodromy group is almost always Sn or An. Another application concerns subgroup growth. We show that a Fuchsian group Γ has (n!)µ+o(1) index n subgroups, where µ is the mea- sure of Γ, and derive similar estimates for so-called Eisenstein numbers of coverings of Riemann surfaces.