# MA648 Hyperbolic Geometry

**Lecturer: Adam Epstein**

**Term(s):** Term 2

**Status for Mathematics students:** List C

**Commitment:** 30 Lectures

**Assessment:** 3-hour examination, 100%.

**Formal registration prerequisites: **None

**Assumed knowledge:**

- MA259 Multivariable Calculus
- General notions of metric, topology, and continuity as presented in MA260 Norms, Metrics and Topologies or MA222 Metric Spaces

**Useful background:**

**Synergies:**

**Leads To: **

**Content**: An introduction to hyperbolic geometry, mainly in dimension two, with emphasis on concrete geometrical examples and how to calculate them. Topics include: basic models of hyperbolic space; linear fractional transformations and isometries; discrete groups of isometries (Fuchsian groups); tesselations; generators, relations and PoincarĂ©'s theorem on fundamental polygons; hyperbolic structures on surfaces.

**Aims**: To introduce the beautiful interplay between geometry, algebra and analysis which is involved in a detailed study of the PoincarĂ© model of two-dimensional hyperbolic geometry.

**Objectives**: To understand:

- The non-Euclidean geometry of hyperbolic space
- Tesselations and groups of symmetries of hyperbolic space
- Hyperbolic geometry on surfaces

**Books**:

J.W. Anderson, *Hyperbolic Geometry*, Springer Undergraduate Math. Series.

S. Katok, *Fuchsian Groups,* Chicago University Press.

S. Stahl, *The PoincarĂ© Half-Plane*, Jones and Bartlett.

A. Beardon, *Geometry of Discrete Groups*, Springer.

J. Lehner, *Discontinuous Groups and Automorphic Functions*. AMS.

L. Ford, *Automorphic Functions*, Chelsea (out of print but in library).

J. Stillwell, *Mathematics and its History*, Springer.