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The Arithmetic of Kleinian Groups April 20, 2016 Contents 1 Overview3 1.1 Course Description................................3 1.2 Pre-requisites...................................3 1.3 Resources.....................................3 1.4 Credit.......................................4 2 Matrix groups, General and Special Linear Groups5 2.1 Topological groups................................5 2.2 Linear Groups...................................5 2.3 Discrete groups..................................7 3 Complex M¨obiustransformations9 1 3.1 P (C) and the chordal metric..........................9 3.2 Circles and circle inversions in Cb ........................ 10 3.3 M¨obiustransformations............................. 13 3.4 Conformality and orientation.......................... 16 3.5 The Action of GL2(C) and PGL2(C)...................... 18 3.6 Cross-ratio.................................... 19 3.7 The M¨obius action on circles.......................... 21 3.8 The inversive pairing on circles......................... 23 3.9 Circles as elements of a Grassmanian...................... 24 3.10 Topological action................................ 26 3.11 Conjugacy classes: trace and fixed points................... 29 4 Hyperbolic geometry 34 4.1 Quaternion representation and Poincar´eextension.............. 34 4.2 Hermitian forms................................. 36 4.3 Hyperbolic functions............................... 37 4.4 The upper half plane model of the hyperbolic plane.............. 38 4.5 Isometries of the upper half plane........................ 40 4.6 Geometry of H2 ................................. 42 4.7 Quaternionic Hermitian forms.......................... 45 4.8 Hyperbolic 3-space (upper half space model)................. 46 4.9 Isometries of the Upper Half Space....................... 48 4.10 Geometry of H3 ................................. 50 1 4.11 The Klein models................................. 53 4.12 Hyperbolic geometry in the plane: area, triangles, etc............. 55 5 Subgroups of PSL2(C) 58 5.1 Fixed points and stabilizers for PSL2(C).................... 58 5.2 Point stabilizers.................................. 61 5.3 `Small' subgroups of PSL2(C)......................... 63 6 Discontinuous action and fundamental domains 65 6.1 Discontinuous groups............................... 65 6.2 Fundamental domains.............................. 67 6.3 Limit sets and regular sets............................ 74 7 Hyperbolic Manifolds and Orbifolds 79 7.1 Hyperbolic manifolds and fundamental groups................. 79 7.2 Dirichlet domains, Cayley graphs, and the fundamental group........ 80 7.3 Ends........................................ 81 7.4 Elementary discrete stabilizers......................... 81 7.5 Orbifolds and Selberg's Lemma......................... 85 7.6 Fuchsian and Schottky groups.......................... 87 7.7 Figure-Eight Knot Complement......................... 88 7.8 Bianchi groups and fundamental domains................... 89 7.9 Poincar´e'sPolyhedron Theorem......................... 91 8 Review of Algebraic Number Theory 93 9 Quaternion algebras 93 9.1 Important Warning................................ 93 9.2 Algebras...................................... 93 9.3 The Hamilton Quaternions........................... 94 9.4 Hilbert symbol.................................. 95 9.5 Centrality and Simplicity............................ 97 9.6 Brauer Groups, Wedderburn and Skolem Noether { in brief......... 97 9.7 Characterising Quaternion Algebras...................... 98 9.8 Orders in Quaternion Algebras......................... 99 9.9 Orders in M2(k)................................. 105 9.10 Quaternion Algebras and Quadratic Forms.................. 107 9.11 Quaternion Algebras over local fields...................... 110 9.12 Quaternion algebras over number fields..................... 112 2 10 Trace Fields and Quaternion Algebras for Kleinian Groups 113 10.1 Trace Fields.................................... 113 10.2 Quaternion Algebras............................... 115 10.3 Invariant trace field and invariant quaternion algebra............. 116 10.4 The Trace..................................... 118 10.5 Examples..................................... 120 11 Applications and places to go 121 11.1 Discreteness criterion............................... 121 12 Errata 121 12.1 Maclachlan and Reid............................... 121 1 Overview 1.1 Course Description A Kleinian group is a discrete subgroup of PSL2(C). Examples include 1. PSL2(Z), 2. the Apollonian group, whose limit set is shown here: 3. the Bianchi groups PSL2(OK ) where OK is the ring of integers of an imaginary quadratic field K. There are arithmetic invariants associated to a Kleinian group, specifically, the invariant trace field and invariant quaternion algebra. The purpose of the course is to the study the relationship between these arithmetic invariants and the geometry of Kleinian groups. In order to study this, we will first need to study quaternion algebras, M¨obius transformations, and hyperbolic geometry. 1.2 Pre-requisites I will assume a solid working knowledge of algebraic number theory, e.g. as in Samuel's Algebraic Theory of Numbers and Chapter 0 of Maclachlan and Reid, including basic local theory. I will also assume knowledge of other graduate pillar topics (topology, analysis) as needed. 1.3 Resources 1. Maclachlan and Reid, The arithmetic of hyperbolic 3-manifolds. This is the main resource for the latter portion of the course. Chapter 0 will be assumed. 3 2. Beardon, The geometry of discrete groups. This is the main resource for the back- ground in hyperbolic geometry and Kleinian groups in the first portion of the course. Chapter 1 will be assumed. 3. Samuel, Algebraic theory of numbers. This covers the necessary algebraic number theory background. It will be assumed. 4. Marden, Outer circles: an introduction to hyperbolic 3-manifolds. This introduces M¨obiustransformations and Kleinian groups and their relationship to hyperbolic 3- manifolds. It also has lots of `explorations' that are suitable for further work for credit. It is good for an overview, but lacks details. 5. Parker, Hyperbolic Spaces: The Jyv¨askyl¨aNotes. Available at maths.dur.ac.uk/ ~dma0jrp/img/HSjyvaskyla.pdf. This introduces the hyperbolic upper half plane and space in a nice algebraic way. 6. Katok, Fuchsian Groups. This does only the Fuchsian (H2) case, but it is pleasantly compact. 1.4 Credit Students wishing to receive full credit (an A) for the course shall 1. attend lecture regularly, and 2. demonstrate engagement with the material. Demonstrating engagement will be in the form of one of the following: (a) Demonstrating evidence of significant homework exercises (suggested exercises will be provided), numbering at least 15. These needn't be distributed through- out the full course, allowing students to concentrate on one topic (e.g. M¨obius transformations or quaternion algebras) to mastery, if desired. However, stu- dents should respect their own time sufficiently so that they choose exercises they will learn something useful from. (b) Learning and presenting, in class, a further topic or related material. For ex- ample, one of the examples or applications of Chapters 4 and 5 of Maclachlan and Reid. This will involve producing written notes to be handed out to the class. It should involve at least one lecture, but more is fine if appropriate to the topic. (c) Covering for me while I'm gone at a conference; this will involve digesting and presenting course notes I leave for you. This should involve at least two lectures. (d) Any other agreed-upon demonstration. 4 Students taking the course for credit/no credit are also expected to attend regularly and do something to demonstrate engagement, but it can be smaller in scale. A note on exercises: the ones suggested in the text vary greatly in difficulty, length and scope. Accordingly, the number 15 is only a guide. You may also choose exercises from the books, or make up your own. 2 Matrix groups, General and Special Linear Groups 2.1 Topological groups A group G which is also a topological space, and such that its multiplication G × G ! G; (x; y) 7! xy and its inverse G ! G; x 7! x−1 are continuous maps, is called a topological group. Exercise 1. A subgroup of a topological group is a topological group under the subspace topology. Exercise 2. Let g 2 G, where G is a topological group. Multiplication by g is continuous. Exercise 3. Give an example of a group G, and a topology on it which makes multiplication continuous but under which the inverse fails to be continuous. 2.2 Linear Groups Given an integral domain R (generally, a subring of C), we define a c M (R) = ; a; b; c; d 2 R ; 2 b d a c GL (R) = ; a; b; c; d 2 R; ad − bc 2 R∗ ; 2 b d and also a c SL (R) = ; a; b; c; d 2 R; ad − bc = 1 : 2 b d Importantly, for R = C, these last two are topological groups under matrix multiplica- tion and an appropriate topology. We will define a metric on M2(C), given by the bilinear t product of the trace pairing. Let A∗ denote the conjugate transpose of A, i.e. A . Definition 4. Let A; B 2 M2(C). Define [A; B] = tr(AB∗); called the trace pairing. 5 Proposition 5. The trace pairing is a Hermitian form on the complex vector space M2(C). 1=2 Furthermore, it induces a norm jjAjj = [A; A] , which in turn induces a metric on M2(C), and hence a topology. Furthermore, we have some relationships with the metric on C and matrix multiplica- tion: 1. j det Aj · jjA−1jj = jjAjj 2. j[A; B]j ≤ jjAjj · jjBjj 3. jjABjj ≤ jjAjj · jjBjj
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