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Abstract: The idea is to get the audience introduced to looking at groups (finitely generated) as geometric objects and give some illustration as to how the geometry of a group intricately shapes its algebraic properties. The goal of the lecture series will be to prove the following theorem : If a group G is quasi-isometrically equivalent to the group of integers Z (i.e the geometry of G is equivalent to the geometry of the group of integers), then G contains a finite index subgroup isomorphic to Z!

We will revisit some basic group theory like free groups, group presentations. We will learn about Cayley graphs, the metric space associated with a group.

Then we will look at quasi-isometry of metric spaces in detail that essentially captures the "coarse geometry" of a space. We will prove the Milnor-Schwarz lemma, which is known to be the fundamental lemma in Geometric Group Theory that basically says that if a group G acts on a nice metric space X in a nice way, then  G is geometrically equivalent to X (the geometry of a group is determined by the geometry of the space on which it nicely acts!). Finally we will end with the proof of the theorem that says that If a group G is quasi-isometrically equivalent to Z, then G contains a finite index subgroup isomorphic Z! This is an instance of a vast area of research called quasi isometric rigidity.