Introduction to Geometric Group Theory
This is a previous supplementary module, not currently scheduled to run this session.
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Geometric group theory is an extremely active field, whose premise is to understand and exploit the actions of infinite groups on geometric spaces such as graphs, cube complexes, hyperbolic spaces and manifolds, in order to unveil key algebraic, algorithmic or topological properties of the groups. Since groups arise naturally in most parts of mathematics and physics, geometric group theory provides new tools to tackle problems from various research topics, and it has had striking applications to low-dimensional topology and birational geometry, among others.
This course will be an introduction to this dynamic field, and will be structured in three parts. In the first part we will give the necessary background on group presentations and Cayley graphs, group actions on graphs and spaces, and quasi-isometries, culminating with the Švarc-Milnor theorem. We will then illustrate its general philosophy through the study of two important classes of groups at the crossroads between several areas: in part two we will focus on hyperbolic groups and hyperbolic geometry, and in part three we will present reflection and Coxeter groups through their actions. In each case, students will see how the geometric tools allow them to obtain new algebraic information about the groups, such as solving the word problem in hyperbolic groups, or understanding the subgroups of Coxeter groups.