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, Jorgensen inequality.
Nov 12
Fundamental domains
Drichlet domain, Modular group, Locally finite domain.
Nov 19
A work of Siegel
Some remarks on discontinuous groups, The Annals of Mathematics, Second Series, Vol. 46, No. 4, (Oct., 1945), pp. 708-718.
Nov 26

Either study signature of a Fuchsian group, or reserve this time to catch up with the schedule.
Dec 3
Continued fraction
C. Series, The modular surface and continued fractions, J. London Math. Soc. (2), 31 (1985), 69-80.
Dec 10
Uniformization theorem
Hyperbolic surface, Hopf-Rinow theorem, Uniformization theorem. (HG')
Dec 17
Monodromy theorem
Geodesic lifting property, Monodromy theorem. (HG')