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´e disc 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
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                  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
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TheUniversity 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: charles.walkden@manchester.ac.uk.
                      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
                   Onsuccessfully 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.
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TheUniversity of Manchester 2019                                                 3