Junior Seminar: Hyperbolic geometry

Fall 2008

Tuesday 5:30-
601 Fine


  • 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.
  • S. Katok, Fuchsian groups. We will more or less cover the first four chapters of this book.
  • B. Iversen, Hyperbolic geometry. Unfortunately it is out of stock. 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.

Sep 23
Fundamental concepts
Parallel postulate. Different models. Hyperbolic metric.
Sep 30
Hyperbolic area and trigonometry
Gauss-Bonnet, Angle of parallelism, The sine rule, The cosine rule I, II.
Oct 7
Area of a polygon, Convex polygon, Quadrilaterals, Pentagons, Hexagons.
Oct 14
The geometry of geodesics
Distance from a line, Perpendicular bisector, Common orthogonal of disjoint geodesics, Pencils of geodesics.
Oct 21
Geometry of isometries
Classification of isometries, Displacement function, Canonical region.
Nov 4
Fuchsian groups
Discreteness criteria (GDG) or (HG'), Algebraic properties (FG), Elementary Fuchsian groups (FG), Jorgensen inequality (FG).
Nov 11
Fundamental domains
Drichlet domain, Modular group, Locally finite domain.
Nov 18
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 25

Either study signature of a Fuchsian group, or reserve this time to catch up with the schedule.
Dec 2
Continued fraction
C. Series, The modular surface and continued fractions, J. London Math. Soc. (2), 31 (1985), 69-80.
Dec 9
Uniformization theorem
Hyperbolic surface, Hopf-Rinow theorem, Uniformization theorem. (HG')
Jan ?
Monodromy theorem
Geodesic lifting property, Monodromy theorem. (HG')