# MA648 Hyperbolic Geometry

**Lecturer: Dr. Salim Ghazouani**

**Term(s):** Term 2

**Status for Mathematics students:** List C

**Commitment:** 30 Lectures

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

**Prerequisites:** MA225 Differentiation and MA3F1 Introduction to Topology. MA3B8 Complex Analysis strongly recommended. Closely related to Geometric Group Theory MA4H4, MA475 Riemann surfaces, MA455 Manifolds

**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 Poincaré'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 Poincaré 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 Poincaré 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.