General Information:
Instructor: Bjoern MuetzelCourse description:
By associating a group to one of its Cayley graphs, the properties of a
group can be studied from a geometric point of view. The group itself
acts on the Cayley graph via isometries which is reflected in the
symmetries of the graph. The inherent beauty of a group can thus be
visualized in its Cayley graph making these graphs a fascinating
object of study.
Geometric group theory examines the connection
between geometric and algebraic invariants of a group. In order to
obtain interesting invariants one usually restricts oneself to
finitely generated groups and takes invariants from large scale
geometry.
Geometric group theory closely interacts with
low-dimensional topology, hyperbolic geometry and differential
geometry and has numerous applications to problems in classical
fields, like combinatorial group theory, graph theory
and differential topology.
Topics: Graphs and trees, Cayley graphs, free groups, hyperbolic groups, large scale geometry.
Contents:
Chapter I - Graphs and trees
Chapter II - Cayley graphs
Chapter III - Geometric realizations of graphs
Chapter IV - Finitely generated groups
Chapter V - Geometry from far away
Chapter VI - Space of ends
Chapter VII - Hyperbolic groups
Homework:
Homework problems will be assigned weekly and discussed in the next problem session. Collaboration on homework is permitted and encouraged. Any resource is allowed provided you reference it. But you must write up your solutions by yourself.Special considerations:
Students with disabilities who will be taking this course and may need disability-related classroom accommodations are encouraged to make an appointment to see their instructor as soon as possible.Literature:
- Brian Bowditch: A course on geometric group theory, lecture notesBackground reading:
- Bridson, Haefliger: Metric spaces of non-positive curvature