Teaching
I taught a course on geometric group theory in the winter semester of 2025/6, the notes for which are can be found here. Comments and corrections are always appreciated. Below you can find a breakdown of the notes in terms of what I covered in each lecture, which may differ in some respects from the above document.
Notes by lecture
- Lecture 1: Group presentations, actions, Cayley graphs, and quasi-isometries.
- Lecture 2: Quasi-geodesics, the Milnor-Schwarz lemma, classical hyperbolic geometry.
- Lecture 3: Some more classical hyperbolic geometry, hyperbolic metric spaces, the Morse Lemma.
- Lecture 4: Quasi-isometric invariance of hyperbolicity, local quasigeodesics in hyperbolic spaces, quasiconvexity.
- Lecture 5: The boundary of a hyperbolic metric space, isometries of hyperbolic metric spaces.
- Lecture 6: Hyperbolic groups, centralisers of elements, and finite subgroups.
- Lecture 7: Convergence groups: set-up and basic theory.
- Lecture 8: Ping-pong, Tits' alternative, conical limit points and uniform convergence groups.
- Lecture 9: Topological refomulation of convergence groups, groups acting on hyperbolic spaces are convergence.
- Lecture 10: Boundaries of hyperbolic groups: topology and classification.
- Lecture 11: Algorithms and group theory, small cancellation theory.
- Lecture 12: The Rips construction, quasiconvex subgroups, and classifying spaces for hyperbolic groups.
- Lecture 13: Amalgamated products, HNN extensions, ends of groups and Stallings' theorem.
- Lecture 14: Bass--Serre theory: theory, examples, and applications, accessibility theory.