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# BSc Actuarial Science and Mathematics / Course details

Year of entry: 2021

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## Course unit details:Hyperbolic Geometry

Unit code MATH32051 10 Level 3 Semester 1 Department of Mathematics No

### 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.

### Pre/co-requisites

Unit title Unit code Requirement type Description
Foundations of Pure Mathematics A MATH10101 Pre-Requisite Compulsory
Foundations of Pure Mathematics B MATH10111 Pre-Requisite Compulsory
Calculus and Vectors A MATH10121 Pre-Requisite Compulsory
Calculus and Vectors B MATH10131 Pre-Requisite Compulsory

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.

### Learning outcomes

On successfully completing the course students will be able to:

• calculate the hyperbolic distance between and the geodesic through points in the hyperbolic plane,
• compare different models (the upper half-plane model and the Poincaré disc model) of hyperbolic geometry,
• prove results (Gauss-Bonnet Theorem, angle formulæ for triangles, etc as listed in the syllabus) in hyperbolic trigonometry and use them to calculate angles, side lengths, hyperbolic areas, etc, of hyperbolic triangles and polygons,
• classify Möbius transformations in terms of their actions on the hyperbolic plane,
• calculate a fundamental domain and a set of side-pairing transformations for a given Fuchsian group,
• define a finitely presented group in terms of generators and relations,
• use Poincaré’s Theorem to construct examples of Fuchsian groups and calculate presentations in terms of generators and relations,
• relate the signature of a Fuchsian group to the algebraic and geometric properties of the Fuchsian group and to the geometry of the corresponding hyperbolic surface.

### 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.

### Assessment methods

Method Weight
Other 20%
Written exam 80%

Coursework test to be held in Week 6, weighting within unit 20%

End of semester examination; weighting within unit 80%.

### 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.

• 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.

### Study hours

Scheduled activity hours
Lectures 22
Tutorials 11
Independent study hours
Independent study 67

### Teaching staff

Staff member Role
Charles Walkden Unit coordinator