Hyperbolic Geometry

 Unit code: MATH42051 Credit Rating: 15 Unit level: Level 4 Teaching period(s): Semester 1 Offered by School of Mathematics Available as a free choice unit?: N

Requisites

Prerequisite

MATH42051 pre-requisites

It is strongly recommended that Year 3 students choosing this course have marks of 55% in MATH20122 Metric Spaces.

Students are not permitted to take more than one of MATH32051 or MATH42051 for credit in the same or different undergraduate year.  Students are not permitted to take MATH42051 and MATH62051 for credit in an undergraduate programme and then a postgraduate programme.

Aims

To provide an introduction to the hyperbolic plane and hyperbolic geometry. To study how discrete groups of isometries act on the hyperbolic plane.

Overview

Consider the Euclidean plane R2. If we take a straight line L and a point p not on that line, then there is a unique straight line through p that never intersects L (draw a picture!). This is Euclid's parallel postulate. Euclid introduced several axioms for what is now called Euclidean geometry (that is, geometry in R2 or more generally inRn and a great deal of effort was employed in attempting to prove that these axioms implied the parallel postulate. However, in early 19th century, the hyperbolic plane was introduced as a setting in which Euclid's axioms hold but the parallel postulate fails: there may be infinitely many "straight" lines through a point that do not intersect a given "straight" line.

Today, hyperbolic geometry is a rich and active area of mathematics with many beautiful theorems (and can be used to generate very attractive pictures)

This course provides an introduction to hyperbolic geometry. We start by discussing what is meant by "distance" and what is "straight" about a straight line in the Euclidean plane R2. We then give an introduction to the hyperbolic plane. Topics include: distance and area in the hyperbolic plane, distance-preserving maps, hyperbolic trigonometry and hyperbolic polygons.

The collection of all distance-preserving maps forms a group. The second part of the course studies a particular class of such groups, namely Fuchsian groups. By using a very beautiful theorem called Poincaré's Theorem, we will describe the connections between such groups and tessellations (tilings) of the hyperbolic plane. The emphasis here will be on how to calculate with and apply Poincaré's Theorem, rather than on rigorous proofs.

One aim of the course is to show how results and techniques from different areas of mathematics, notably geometry, algebra and analysis, can be used coherently in the study of a single topic.

Learning outcomes

On successfully completing the course students will be able to:

• understand the geometry of the hyperbolic plane and be able to prove results and identities in hyperbolic geometry and trigonometry,
• understand other models of the hyperbolic plane, such as the Poincaré disc model,
• understand how the group of Möbius transformations acts on the hyperbolic plane by isometries,
• understand how discrete groups of isometries act on the hyperbolic plane and their connections with fundamental domains and Dirichlet regions

Assessment methods

• Other - 7%
• Written exam - 93%

Assessment Further Information

Coursework: Multiple choice test held during Week 6; weighting within unit 7%.

Two hour end of semester examination; weighting within unit 93%.

Syllabus

• Introduction, background and motivation.
• The upper half-plane model, hyperbolic distance and area, geodesics. The group of Möbius transformations as isometries.
• The Poincaré disc model. Möbius transformations of the Poincaré disc.
• Hyperbolic triangles, hyperbolic trigonometry, hyperbolic polygons
• Classifying different types of isometries.
• Introduction to discrete groups of isometries.
• Fundamental domains and Dirichlet regions.
• Poincaré's theorem and groups generated by side-pairing transformations.
• Limit sets of Fuchsian groups (independent reading material).

• J. Anderson, Hyperbolic Geometry, Springer, 1999.
• S. Katok, Fuchsian Groups, Chicago, 1992
• A. Beardon, The Geometry of Discrete Groups, Springer, 1983

The book by Anderson is the most suitable for the course.

Feedback methods

Feedback tutorials will provide an opportunity for students' work to be discussed and provide feedback on their understanding.  Coursework or in-class tests (where applicable) also provide an opportunity for students to receive feedback.  Students can also get feedback on their understanding directly from the lecturer, for example during the lecturer's office hour.

Study hours

• Lectures - 22 hours
• Tutorials - 16 hours
• Independent study hours - 112 hours

Teaching staff

Charles Walkden - Unit coordinator