Seminar on Hyperbolic Geometry, Winter Semester 2025/26

Prof. Dr. Joachim Lohkamp, Dr. Matthias Kemper

Hyperbolic geometry originated in the 19th century, when mathematicians questioned the necessity of the parallel postulate in Euclidean geometry and discovered the hyperbolic plane ℍ², which satisfied all of Euclid’s axioms except for the parallel postulate. Later, the term “hyperbolic” was applied to all kinds of spaces sharing some features with the hyperbolic plane, such as:

higher-dimensional hyperbolic spaces ℍⁿ,
groups acting isometrically on ℍⁿ,
the resulting quotients, Riemannian manifolds of sectional curvature −1,
metric spaces with triangles slimmer than those in ℍ², called CAT(−1) spaces,
and metric spaces with intrinsically thin triangles, where every side of a triangle is contained in a δ-neighbourhood of the other two sides, for some universal δ > 0, called δ-hyperbolic spaces.

It turns out that these conditions tend to both occur frequently and be helpful for further analysis. For example, all genus > 1 compact surfaces admit a hyperbolic metric, while in dimension ≥ 3, the Mostow rigidity theorem holds: Isometries of finite-volume hyperbolic manifolds are completely determined by isomorphisms of their fundamental group.
In group theory, in a certain sense, “almost all” finitely presentable groups are δ-hyperbolic, and hyperbolicity enables an efficient algorithm to solve the word problem.

In this seminar, we will explore these and more hyperbolic phenomena, mostly from a geometric perspective. Presentation topics can be selected according to the interests and previous knowledge of the participants. There will be topics suitable for both bachelor and master students. Some familiarity with Riemannian manifolds is helpful, but not strictly necessary for every topic.

There will be an introductory meeting at the beginning of the semester.

Please register in Learnweb if you are interested in participating, then we will notify you.