104.741 — Group Actions on Hyperbolic Spaces

AKALG Gruppenwirkungen auf hyperbolischen Räumen

Lecturer

Chris Cashen

Schedule

Fridays, 14:00–17:00, 9 October 2026 – 22 January 2027

Room: Sem.R. DA grün 06A (SEM 107)

Course information

• Weekly semester hours: 3.0
• ECTS: 4.0
• Type: Lecture with exercises
• Format: In person

Learning outcomes

Upon successful completion of the course, students will be able to describe examples of hyperbolic spaces and their isometry groups; explain Gromov hyperbolicity and verify it in standard examples; analyze group actions using orbits and stabilizers; classify isometries; apply the Milnor–Švarc lemma; use boundaries of hyperbolic spaces; and compare hyperbolic, relatively hyperbolic, and acylindrically hyperbolic group actions.

Course content

This course provides an introduction to group actions on hyperbolic spaces, with a focus on geometric group theory. We begin with rank-one symmetric spaces, particularly real hyperbolic space, and discuss their isometry groups and basic geometric features. We then examine Gromov hyperbolic spaces and hyperbolic groups, including quasi-isometries, the Milnor–Švarc lemma, the stability of quasigeodesics, boundaries, and examples. The second part of the course introduces relatively hyperbolic groups and their actions on cusped and coned-off spaces. Finally, we discuss acylindrically hyperbolic groups, loxodromic elements, and examples arising from mapping class groups, outer automorphism groups of free groups, and groups acting on CAT(0) cube complexes.

Methods

The course is taught through blackboard lectures and joint problem-solving sessions.

Lecture Notes

Further reading

• Cornelia Druţu and Michael Kapovich, Geometric Group Theory (AMS, 2018). Free author version (PDF). A comprehensive reference.
• Clara Löh, Geometric Group Theory: An Introduction (Springer, 2017). Free draft (PDF). A more accessible introduction.
• Martin R. Bridson and André Haefliger, Metric Spaces of Non-Positive Curvature (Springer, 1999). Another comprehensive reference.
• Matt Clay and Dan Margalit (eds.), Office Hours with a Geometric Group Theorist, Princeton University Press, 2017.
• Pierre de la Harpe, Topics in Geometric Group Theory (University of Chicago Press, 2000).
• Brian H. Bowditch, A Course on Geometric Group Theory. Introductory notes covering groups, geometry, and hyperbolicity.
• Michael Hull, Hyperbolic Groups. Notes on hyperbolic spaces and groups.
• Brian H. Bowditch, Relatively Hyperbolic Groups.
• Denis Osin, Groups Acting Acylindrically on Hyperbolic Spaces.