Math 503. Introduction to Geometric Group Theory Instructor Syllabus
Geometric Group Theory is an actively developing area of mathematics drawing on the ideas and techniques from Riemannian geometry, lowdimensional topology, combinatorics, analysis, probability, logic as well as the traditional group theory. A key underlying idea of Geometric Group Theory is to study the interaction between algebraic properties of a finitely generated group and geometric properties of a space admitting a “nice” isometric action of this group.
- Gilbert Baumslag “Topics in Combinatorial Group Theory”, Birkhüser Verlag, Basel, 1993.
- Pierre de la Harpe "Topics in geometric group theory. Chicago Lectures in Mathematics". University of Chicago Press, Chicago, IL, 2000.
(1) Free groups and their subgroups via Stallings subgroup graphs.
(2) Groups given by generators and relators. Cayley graphs and the word metric. Van Kampen diagrams and van Kampen Theorem. Time permitting, a discussion of small cancellation theory.
(3) Bass-Serre theory: Amalgamated free products and HNN-extensions, graphs of groups and group actions on simplicial trees.
(4) Derived series, upper and lower central series, nilpotent and solvable groups, commutator calculus. Finitely generated and finite nilpotent groups. Semi-direct products and wreath products.
(5) Quasi-isometries, Milnor-Swarc theorem, geometric properties and invariants. Examples of quasi-isometric invariants: ends, growth, isoperimetric functions, amenability, solvability of the word problem and asymptotic cones.
(6) A sample advanced topic, e.g. one or more of the following: wordhyperbolic groups; the Novikov-Boone Theorem; the Higman Embedding Theorem; Burnside groups; Grigorchuk groups of intermediate growth; automatic groups; relative hyperbolicity; actions on R-trees, etc.