General Prerequisites: Knowledge of the material in the courses B3.5 Topology and Groups<\i> and C2.4 Infinite Groups<\i> is helpful but not mandatory, as it would only bring more familiarity and a different viewpoint of some of the notions treated in this course.
Course Overview: The aim of this course is to introduce fundamental methods of geometric group theory, used to investigate infinite groups. It focuses on ``large groups'', that is, groups that are similar to free non-abelian groups. The two main methods used are the study of actions of such groups on metric spaces, and the investigation of some algebraic problems by designing appropriate algorithms.
The course begins with an introduction to presentations and the list of problems of Max Dehn. It continues with the study of actions of groups on trees and the decomposition of infinite groups into simpler building blocks that such an action can yield. It provides a structural study of fundamental groups of graphs of groups.
The second part of the course focuses on modern geometric techniques and introduces a few key results from the theory of Gromov hyperbolic groups.
Course Synopsis: Free groups. Group presentations. Dehn's problems. Residually finite groups.
Group actions on trees. Amalgams, HNN-extensions, graphs of groups, subgroup theorems for groups acting on trees.
Quasi-isometries. Hyperbolic groups. Solution of the word and conjugacy problem for hyperbolic groups.
If time allows: Small Cancellation Groups, Stallings Theorem, Boundaries.
