MATH32052 Hyperbolic Geometry Charles Walkden 12th January, 2019 MATH32052 Contents Contents 0 Preliminaries 3 1 Where we are going 6 2 Length and distance in hyperbolic geometry 13 3 Circles and lines, M¨obius transformations 18 4 M¨obius transformations and geodesics in H 23 5 More on the geodesics in H 26 6 The Poincar´edisc model 39 7 The Gauss-Bonnet Theorem 44 8 Hyperbolic triangles 52 9 Fixed points of M¨obius transformations 56 10 Classifying M¨obius transformations: conjugacy, trace, and applications to parabolic transformations 59 11 Classifying M¨obius transformations: hyperbolic and elliptic transforma- tions 62 12 Fuchsian groups 66 13 Fundamental domains 71 14 Dirichlet polygons: the construction 75 15 Dirichlet polygons: examples 79 16 Side-pairing transformations 84 17 Elliptic cycles 87 18 Generators and relations 92 19 Poincar´e’s Theorem: the case of no boundary vertices 97 20 Poincar´e’s Theorem: the case of boundary vertices 102 c The University of Manchester 1 MATH32052 Contents 21 The signature of a Fuchsian group 109 22 Existence of a Fuchsian group with a given signature 117 23 Where we could go next 123 24 All of the exercises 126 25 Solutions 138 c The University of Manchester 2 MATH32052 0. Preliminaries 0. Preliminaries 0.1 Contact details § The lecturer is Dr Charles Walkden, Room 2.241, Tel: 0161 27 55805, Email: [email protected]. My office hour is: WHEN?. If you want to see me at another time then please email me first to arrange a mutually convenient time. 0.2 Course structure § 0.2.1 MATH32052 § MATH32052 Hyperbolic Geoemtry is a 10 credit course. There will be about 22 lectures and a weekly examples class. The examples classes will start in Week 2. 0.2.2 Learning outcomes § On successfully completing the course you will be able to: ILO1 calculate the hyperbolic distance between and the geodesic through points in the hyperbolic plane, ILO2 compare different models (the upper half-plane model and the Poincar´e disc model) of hyperbolic geometry, ILO3 prove results (Gauss-Bonnet Theorem, angle formulæ for triangles, etc as listed in the syllabus) in hyperbolic trigonometry and use them to calcu- late angles, side lengths, hyperbolic areas, etc, of hyperbolic triangles and polygons, ILO4 classify M¨obius transformations in terms of their actions on the hyperbolic plane, ILO5 calculate a fundamental domain and a set of side-pairing transformations for a given Fuchsian group, ILO6 define a finitely presented group in terms of generators and relations, ILO7 use Poincar´e’s Theorem to construct examples of Fuchsian groups and calculate presentations in terms of generators and relations, ILO8 relate the signature of a Fuchsian group to the algebraic and geometric properties of the Fuchsian group and to the geometry of the corresponding hyperbolic surface. c The University of Manchester 2019 3 MATH32052 0. Preliminaries 0.2.3 Lecture notes § This file contains a complete set of lecture notes. The lecture notes contain more material than I present in the lectures. This allows me to expand on minor points for the interested student, present alternative explanations, etc. Only the material I cover in the lectures is examinable. The lecture notes are available on the course webpage. The course webpage is available via Blackboard or directly at www.maths.manchester.ac.uk/~cwalkden/complex-analysis. Please let me know of any mistakes or typos that you find in the notes. I will use the visualiser for the majority of the lectures. I will upload scanned copies of what I write on the visualiser onto the course webpage. I will normally upload these onto the course webpage within 3 working days of the lectures. 0.2.4 Exercises § The lectures also contain the exercises. For your convenience I’ve collated all the exercises into a single section at the end of the notes; here, I’ve indicated which exercises are partic- ularly important and which are there for completeness only. The exercises are a key part of the course. 0.2.5 Solutions to the exercises § This file contains the solutions to all of the exercises. I will trust you to have a serious attempt at the exercises before you refer to the solutions. 0.2.6 Support classes § The support classes are a key part of the course. I will try to make them as interactive as possible by getting you to revise material that will be useful in the course or getting you to work through some of the exercises, perhaps with additional hints. Towards the end of the course we will spend time doing past exam papers. You should consider attendance at the examples classes to be compulsory. The handouts in the support classes comprise of questions from the exercises and past exams. These handouts do not contain any material that is not already available within these notes or within the past exam papers on the course webpage; as such I will not be putting these handouts on the course webpage. 0.2.7 Lecture capture § The lectures will be recorded using the University’s lecture capture (‘podcast’) system. I will mostly use the visualiser. Each week, I will scan copies of the visualiser slides and put them on the course webpage. 0.3 Coursework and the exam § The coursework for this year will be a 40 minute closed-book test taking place during WHEN???? All questions on the test are compulsory and it will be in the format of an exam question. Thus, looking at past exam papers will provide excellent preparation for the test. You will need to know the material from sections 1–1 in the lecture notes for the test (this is the material that we will cover in weeks 1–5). c The University of Manchester 2019 4 MATH32052 0. Preliminaries Your coursework script, with feedback, will be returned to you within 15 working days of the test. You will be able to collect your script from the Teaching & Learning Office reception on the ground floor of the Alan Turing Building. The course is examined by a 2 hour written examination in May/June. The format of the exam will change this year: there will be 4 questions of which you have to answer 3. (If you attempt all 4 questions then only your best 3 answers will count.) Past exam papers still give a good guide as to what you could be asked to do. 0.4 Recommended texts § J. Anderson, Hyperbolic Geometry, 1st ed., Springer Undergraduate Mathematics Series, Springer-Verlag, Berlin, New York, 1999. S. Katok, Fuchsian Groups, Chicago Lecture Notes in Mathematics, Chicago University Press, 1992. A. Beardon, The Geometry of Discrete Groups, Springer-Verlag, Berlin, New York, 1983. The book by Anderson is the most suitable for the first half of the course. Katok’s book is probably the best source for the second half of the course. Beardon’s book contains everything in the course, and much more. You probably do not need to buy any book and can rely solely on the lecture notes. c The University of Manchester 2019 5 MATH32052 1. Where we are going 1. Where we are going 1.1 Introduction § One purpose of this course is to provide an introduction to some aspects of hyperbolic ge- ometry. Hyperbolic geometry is one of the richest areas of mathematics, with connections not only to geometry but to dynamical systems, chaos theory, number theory, relativity, and many other areas of mathematics and physics. Unfortunately, it would be impossible to discuss all of these aspects of hyperbolic geometry within the confines of a single lecture course. Instead, we will develop hyperbolic geometry in a way that emphasises the similar- ities and (more interestingly!) the many differences with Euclidean geometry (that is, the ‘real-world’ geometry that we are all familiar with). 1.2 Euclidean geometry § Euclidean geometry is the study of geometry in the Euclidean plane R2, or more generally in n-dimensional Euclidean space Rn. This is the geometry that we are familiar with from the real world. For example, in a right-angled triangle the square of the hypotenuse is equal to the sum of the squares of the other two sides; this is Pythagoras’ Theorem. But what makes Euclidean geometry ‘Euclidean’? And what is ‘geometry’ anyway? One convenient meta-definition is due to Felix Klein (1849-1929) in his Erlangen programme (1872), which we paraphrase here: given a set with some structure and a group of trans- formations that preserve that structure, geometry is the study of objects that are invariant under these transformations. For 2-dimensional Euclidean geometry, the set is the plane R2 equipped with the Euclidean distance function (the normal way of defining the distance be- tween two points) together with a group of transformations (such as rotations, translations) that preserve the distance between points. We will define hyperbolic geometry in a similar way: we take a set, define a notion of distance on it, and study the transformations which preserve this distance. 1.3 Distance in the Euclidean plane § Consider the Euclidean plane R2. Take two points x,y R2. What do we mean by the ∈ distance between x and y? If x = (x1,x2) and y = (y1,y2) then one way of calculating the distance between x and y is by using Pythagoras’ Theorem: distance(x,y)= x y = (y x )2 + (y x )2; (1.3.1) k − k 1 − 1 2 − 2 this is the length of the straight line drawn in Figurep 1.3.1. Writing d(x,y) for distance(x,y) we can see that there are some natural properties satisfied by this formula for distance: (i) d(x,y) 0 for all x,y with equality if and only if x = y, ≥ (ii) d(x,y)= d(y,x) for all x,y, c The University of Manchester 2019 6 MATH32052 1.