CONTENTS -- Foreword -- Preface -- An Invitation to Elementary Hyperbolic Geometry Ying Zhang -- Introduction -- 1. Euclid's Elements, Book I and Neutral Plane Geometry -- 1.1. A brief review of contents of Elements, Book I -- 1.2. A useful lemma -- 1.3. A gure-free proof of Proposition I.7 -- 1.4. More notes on Elements, Book I -- 1.5. Playfair's axiom -- 1.6. Neutral plane geometry -- 1.7. Angle-sums of triangles and Legendre's Theorems -- 1.8. Quadrilaterals with two consecutive right angles -- 1.9. Saccheri and Lambert quadrilaterals -- 1.10. Variation of triangles in a neutral plane -- 1.11. A midline configuration for triangles -- 1.12. More theorems of neutral plane geometry -- 1.13. Small angles -- 2. Hyperbolic Plane Geometry -- 2.1. Hyperbolic plane -- 2.2. Asymptotic Parallelism -- 2.3. Angle of parallelism -- 2.4. The variation in the distance between two straight lines -- 2.5. Some more theorems in hyperbolic plane geometry -- 2.6. Construction of the common perpendicular to two ultra-parallel straight lines -- 2.7. Construction of asymptotic parallels -- 2.8. Ideal points -- 2.9. Horocycles -- 2.10. Construction of the straight line joining two given ideal points -- 2.11. Ultra-ideal points -- 2.12. The projective plane associated to a hyperbolic plane -- 2.13. Center-pencils of a hyperbolic triangle -- 2.14. Equidistant curves -- 2.15. Positions of proper points relative to an ideal point -- 2.16. Hyperbolic areas via equivalence of triangles -- 2.17. Metric relations of corresponding arcs in concentric horocycles -- 3. Isometries of the Hyperbolic Plane -- 3.1. Isometries and reections in straight lines -- 3.2. Orientation preserving/reversing isometries -- 3.3. Rotations -- 3.4. Translations -- 3.5. Isometries of parabolic type -- 3.6. Redundancy of two reflections.

3.7. Orientation reversing isometries as reflections and glide reflections -- 3.8. Isometries as projective transformations -- 3.9. Invariant projective lines of -- 3.10. Composition of two orientation preserving isometries other than two translations -- 3.11. Composition of two translations -- 3.12. Conjugates of isometries -- 3.13. The orthic triangle -- 4. Hyperbolic Trigonometry Derived from Isometries -- 4.1. Some identities of isometries of a neutral plane -- 4.2. Some trigonometric formulas in H2(k) -- 4.3. Upper half-plane model U2 for hyperbolic plane H2(1) -- 4.4. Matrices of certain isometries of U2 -- 4.5. Trigonometric laws via identities of isometries -- 4.6. Suggested further readings -- Acknowledgments -- References -- Hyperbolic Structures on Surfaces Javier Aramayona -- 1. Introduction -- 2. Plane Hyperbolic Geometry -- 2.1. Mobius transformations -- 2.1.1. Classification in terms of trace and fixed points -- 2.2. Models for hyperbolic geometry -- 2.2.1. Hyperbolic distance -- 2.2.2. Mobius transformations act by isometries -- 2.2.3. The Cayley transformation -- 2.2.4. Hyperbolic geodesics -- 2.2.5. The boundary at infinity -- 2.2.6. The full isometry group -- 2.2.7. Dynamics of elements of Isom+(H) -- 2.3. Fuchsian groups and fundamental domains -- 2.3.1. Fuchsian groups -- 2.3.2. Fundamental domains -- 2.3.3. The action of a group on a Dirichlet domain -- 3. Hyperbolic Structures on Surfaces -- 3.1. Definition and examples -- 3.2. The Cartan-Hadamard Theorem. Developing map and holonomy -- 3.2.1. The developing map -- 3.2.2. Two technical lemmas -- 3.2.3. Holonomy -- 4. Teichmuller Space -- 4.1. Two definitions -- 4.2. Fenchel-Nielsen coordinates -- 4.2.1. Length functions -- 4.2.2. Multicurves and pants decompositions -- 4.2.3. The Teichmuller space of a pair of pants -- 4.2.4. The coordinates -- 5. Mapping Class Groups.

5.1. Definition and examples -- 5.1.1. Examples of mapping classes -- 5.2. The action of Mod(S) on T (S) -- Acknowledgements -- References -- Degenerations of Hyperbolic Structures on Surfaces Christopher J. Leininger -- 1. Introduction -- 1.1. Notation, terminology and some conventions -- 2. Length Functions -- 3. Measured Foliations -- 3.1. Definition by example -- 3.2. Geometric intersection number I: Measured foliations -- 3.3. Geodesics in the q-metric -- 4. Measured Laminations -- 4.1. Geometric intersection number II: Measured laminations -- 5. Measured Foliations and Measured Laminations -- 6. Dehn-Thurston Coordinates -- 7. Degenerations of Hyperbolic Structures -- 7.1. The fundamental lemma -- 7.2. Final comments on Theorem 7.1 -- 7.2.1. Application to the mapping class group -- 7.2.2. Other approaches -- Acknowledgements -- References -- Ping-Pong Lemmas with Applications to Geometry and Topology Thomas Koberda -- 1. Introduction -- 2. Free Groups and Free Products -- 3. Hyperbolic Geometry -- 4. Right-Angled Artin Groups -- 5. Mapping Class Groups -- 6. Effective Ping-Pong and the Word Problem -- Acknowledgements -- References -- Creating Software for Visualizing Kleinian Groups Yasushi Yamashita -- 1. Introduction -- 2. Python -- 2.1. Python setup -- 2.2. Basic data types and operations -- 2.3. Control ow -- 2.4. Function -- 2.5. Global variable and local variable -- 2.6. GUI -- 2.7. An example -- 3. Once Punctured Torus Groups -- 3.1. The deformation space of once punctured torus groups -- 3.2. Ford domain -- 3.3. J rgensen's normalization -- 3.4. J rgensen's method to construct the Ford domain -- 3.5. Limit set -- 4. OptPy -- 5. Program List -- 6. Using OptPy -- Acknowledgements -- References -- Traces in Complex Hyperbolic Geometry John R. Parker -- 1. Introduction -- 2. Background -- 2.1. Hermitian and unitary matrices.

2.2. Complex hyperbolic space and its isometries -- 3. The Geometry of Isometries -- 3.1. Introduction -- 3.2. Classification of elements of SU(2, 1) by their trace -- 3.3. Traces and eigenvalues for loxodromic maps -- 3.4. Eigenvalues and complex displacement for loxodromic maps -- 4. Two Generator Groups and Fenchel-Nielsen Coordinates -- 4.1. Introduction -- 4.2. Trace identities in M(3, C) -- 4.3. Trace identities in SL(3, C) -- 4.4. Trace parameters for two generator groups of SU(2, 1) -- 4.5. Cross-ratios -- 4.6. Twist-bend parameters -- 5. Traces for Triangle Groups -- 5.1. Introduction -- 5.2. Reflections -- 5.3. Complex reflections in SU(2, 1) -- 5.4. Equilateral triangle groups -- 5.5. General triangle groups -- 5.6. Traces in general triangle groups -- Acknowledgements -- References -- Lorentzian Geometry Todd A. Drumm -- 1. Introduction -- 1.1. Physical considerations -- 2. Flat Lorentz Spaces -- 3. Compactifications -- 3.1. Euclidean compactification -- 3.2. The Einstein Universes -- 3.3. Extending Lorentzian lines in the Einstein Universe -- 3.4. Covers of the Einstein Universe -- 4. Constant Curvature Spaces -- 4.1. Positive curvature -- 4.1.1. The round sphere -- 4.1.2. de Sitter space -- 4.2. Negative curvature: Hyperboloids -- 4.2.1. Hyperbolic space -- 4.2.2. Anti-de Sitter space -- 5. Three Dimensional Flat Lorentz Manifolds -- 5.1. Group actions -- 5.2. Transformations -- 5.3. Margulis invariant -- 5.4. Deformations of hyperbolic surfaces -- 5.5. Crooked planes -- 5.5.1. Domains in Euclidean and hyperbolic spaces -- 5.5.2. Crooked half-spaces -- References -- Connected Components of PGL(2, R)-Representation Spaces of Non-Orientable Surfaces Frederic Palesi -- 1. Introduction -- 2. The Group PSL(2, R) -- 2.1. Universal cover -- 2.2. Classification of elements -- 2.3. The commutator map -- 2.4. The square map.

3. Components for Orientable Surfaces -- 3.1. A topological invariant -- 3.2. Milnor-Wood inequality and a result of Goldman -- 3.3. Surfaces with boundary -- 3.4. Summary of the proof -- 4. Non-Orientable Surfaces -- 4.1. Non-orientable surface group -- 4.2. Second obstruction class -- 4.3. Non-orientable Euler class -- 4.4. Square map -- 4.5. General formula -- Acknowledgements -- References -- Rigidity and Flexibility of Surface Groups in Semisimple Lie Groups Inkang Kim -- 1. Introduction -- 2. Geometric Invariants -- 3. Real Zariski Tangent Space of Character Variety and Local Rigidity -- 4. Surface Group Representation -- 5. Classical Simple Lie Groups and Flexibility -- Acknowledgments -- References -- Abelian and Non-Abelian Cohomology Eugene Z. Xia -- 1. Prelude -- 2. The Representation Variety -- 3. A Reluctant Tour of Category and Groupoid -- 4. Local Functions and Their Derivatives over Rn -- 4.1. Functions -- 4.2. Exterior differentiation -- 4.3. Connections -- 4.4. The gauge group and its action -- 5. Local Functions and Their Derivatives over Cn -- 5.1. Holomorphic functions and forms -- 5.2. The holomorphic structures -- 5.3. The gauge group action I -- 5.4. Holomorphic -connections -- 5.5. The gauge group action II -- 6. Local Examples in Low Dimensions -- 6.1. Connections on an interval -- 6.2. Connections on the unit disk -- 6.3. The holomorphic construction -- 7. Interlude: Manifolds and Functions -- 8. The Cech Construction -- 8.1. Structure sheaves -- 8.2. Sheaves of modules -- 8.3. Sheaf isomorphisms -- 9. The Smooth de Rham Groupoid -- 9.1. Structure sheaf -- 9.2. Tangent, cotangent sheaves and the de Rham complex -- 9.3. Connections -- 9.4. The gauge group and its action -- 10. The Holomorphic de Rham and Dolbeault Groupoid -- 10.1. Holomorphic structures -- 10.2. The gauge group action I -- 10.3. Holomorphic -connections.

11. The Equivalence of Groupoids.