An Introduction to Hyperbolic Geometry

ABOUT THE COURSE: Hyperbolic geometry is a non-Euclidean Geometry which has wide applications in mathematics and is a central object to study group theory from a geometrical view point. Many surfaces and three manifolds exhibit hyperbolic geometry. In this course, we will learn basic hyperbolic geometry and then move on to its applications in geometric group theory.INTENDED AUDIENCE: PG & PhD Students but an UG student with background in topology, algebra and complex analysis can take this course.PREREQUISITES: Topology, Algebra (Group Theory), Complex Analysis

Week 1-5 : Hyperbolic Geometry : Models of Hyperbolic space: Upper half plane and Unit Disc with Poincaré metric, Hyperbolic Inner Product, Geodesics, Isometry Groups, Classification of Isometries, Area of Triangles, Trignometric Identities.
Week 6-8 : Properly discontinous action, Cocompact action, Fuchsian Group, Covering Space, Fundamental Region, Tesellation, Algebraic properties of Fuchsian Group, Hyperbolic surfaces, Poincaré’s Theorem (Statement Only) Closed curves and geodesics on Hyperbolic Surfaces .
Week 9-12 : Slim and Thin triangles, Hyperbolic metric space, Exponential Divergence, Stability of Quasi-geodesics. Hyperbolic groups and its properties.