Junior Seminar: Hyperbolic geometry
Fall 2009

Thursday 6:30-
401 Fine
Books

  • J. W. Anderson, Hyperbolic geometry. Easy treatment of the plane hyperbolic geometry. It covers around half of the topics of this seminar.
  • H. Meschkowski, Non-Euclidean geometry. From histroical and logical point of view. Through parallel postulate. We will discuss these topics only in our first meeting
  • S. Katok, Fuchsian groups. We will more or less cover the first four chapters of this book.
  • B. Iversen, Hyperbolic geometry. I have the library's copy. You can borrow it from me.
  • A. F. Beardon, The geometry of discrete groups. The first five weeks, we more or less follow this book. Chapters 7, 8, and parts of 9, 10 (Clearly, these books overlap.).

DATE
PRESENTER
TOPIC
Sep 24 Alireza Fundamental concepts
Parallel postulate. Different models. Hyperbolic metric. Möbius transformations. Description of the geodesics in the hyper-half plane model.
Oct 1 John Stogin Hyperbolic area and trigonometry
Poincaré dics model, Gauss-Bonnet, The sine rule, The cosine rule I, II. PDF
Oct 8 Alireza Isometries and geometry
Full group of isometries, Distance from a line, Perpendicular bisector, Common orthogonal of disjoint geodesics, hypercycle
Oct 15 David Sprunger Fuchsian groups
Properly discontinuous actions.
Oct 22 David Sprunger

Juan Miguel Ogarrio 		Elementary Fuchsian groups
Classification of isometries, Abelian Fuchsian groups, Classification of elementary Fuchsian groups. PDF
Nov 5 Juan Miguel Ogarrio Non-elementary Fuchsian groups
Discreteness criteria.
Nov 12 Juan Miguel Ogarrio Non-elementary Fuchsian groups
Jorgensen inequality. PDF
Nov 19 Marli Wang Fundamental domains
Modular group. PDF
Dec 3 Alireza Fundamental domains
Drichlet domains, locally finite.
Dec 10 Tim Campion Fundamental domains
Ford fundamental domains, Quotient space. PDF
Dec 17 John Stogin Geometrically finite
Possible covolumes of Fuchsian groups. PDF
Jan 10 Marli Wang Uniformization theorem
Hyperbolic surface, Hopf-Rinow theorem, Uniformization theorem. PDF