Summer minicourse: Boundaries of groups and spaces
This is the homepage for the minicourse I'm teaching from June 1 - June 5, 2020. You can find the lecture notes and exercises from each day of the course below, along with references. You can also find the notes that Casandra Monroe wrote up during lecture.
Course information
There will be an hour of lecture in the morning, followed by another hour-long exercise session in the afternoons.
Lecture time: 10:00am - 11:00am (US central time), Monday 6/1 - Friday 6/5
Lecture location:
Meeting link: https://utexas.zoom.us/j/96382523688.
Meeting ID: 963 8252 3688
Password: [posted in Slack channel]
Exercise sessions: 1:00pm - 2:00pm (US central time)
Exercises
Day 1 (June 1)
Day 2 (June 2)
Day 3 (June 3)
Day 4 (June 4)
Abstract
This course will be an investigation of various kinds of boundaries of spaces that sometimes show up when doing geometric group theory. We'll be focusing on CAT(0) spaces, hyperbolic groups, and relatively hyperbolic groups. We'll define each of these kinds of spaces/groups, and give (at least) one construction for a natural notion of "boundary at infinity" for each of them. The main goal of the course will be to explore the various kinds of structures and properties these boundaries can have, and how they interact with the properties of the space they bound. Specifically, we'll look at things like
- different topologies and metrics on the boundary
- quasi-symmetry (and maybe quasi-conformality)
- quasi-isometric invariance (and lack thereof)
- dynamics of group actions on the boundary
- limit sets and quasiconvex subgroups
This is a lot, so I emphatically do not expect to get through everything on this list.
Course outline
6/1/20
Definition of CAT(0) spaces, the Cartan-Hadamard theorem, and the visual boundary of a CAT(0) space. The cone topology on the visual boundary.
Notes | Cas's notes | Exercises
6/2/20
Visibility of the boundary of a CAT(0) space and embedded flats. CAT(0) groups and the failure of quasi-isometry invariance.
Notes | Cas's notes | Exercises
6/3/20
Quasi-isometries and the Milnor-Schwarz lemma. Delta-hyperbolic spaces, the Morse lemma, and quasi-isometry invariance. The boundary of a delta-hyperbolic space and the cone topology.
Notes | Cas's notes | Exercises
6/4/20
Metrizing the boundary of a hyperbolic space with Gromov products. Cross-ratios on the boundary of a hyperbolic group, and quasimobius maps under quasi-isometries. Triangles in the boundary and the topological characterization of hyperbolic groups.
Notes | Cas's notes | Exercises
6/5/20
Three views of relatively hyperbolic groups: geometrically finite actions, coned-off spaces, and cusped-off spaces. A description of the Bowditch boundary in terms of geodesics.
References
The main reference for the course will be Bridson and Haefliger's Metric Spaces of Non-Positive Curvature. I'll probably also refer to:
- Bowditch, A topological characterization of hyperbolic groups
- Bowditch, Relatively hyperbolic groups
- Croke and Kleiner, Spaces with nonpositive curvature and their ideal boundaries
- Druţu and Kapovich, Geometric Group Theory
- Groves and Manning, Dehn filling in relatively hyperbolic groups
- Paulin, Un groupe hyperbolique est determiné par son bord
- Yaman, A topological characterization of relatively hyperbolic groups