MATS4100 Introduction to Geometric Group Theory (3 cr)
Learning outcomes
鈥ecome familiar with some of the objects/groups studied in Geometric Group Theory: finitely generated groups, Cayley graphs, isometric action on metric spaces.
鈥nderstand geometric actions and their genericity (Svarc-Milnor Lemma)
鈥nderstand some connections between Group Theory and Geometry, e.g. the importance of the fundamental group of a manifold, etc.
鈥earn some of the comparison tools of Coarse Geometry: Lipschitz map, bi-Lipschitz equivalence, quasi-isometric map, quasi-isometry, etc.
鈥ecome familiar with some of the most important theorems of GGT.
Study methods
To pass the course, each student is required to present a problem on the board during the exercise sessions and to take the written exam
Content
鈥undamental group, covering map, universal cover.
鈥roup action by isometries, finitely generated group, Cayley graph.
鈥eometric action, Svarc-Milnor Lemma.
鈥he growth of a group, volume growth of a manifold, Gromov's theorem on polynomial growth.
鈥yperbolic space and manifold, Gromov hyperbolic metric spaces and groups.
鈥menable groups and the Banach Tarski paradox.
Prerequisites
Metric spaces, Algebra 1: Rings and Fields and Algebra 1: Groups