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Dept. of Mathematics, Ramakrishna Mission Vivekananda Educational and Research Institute (RKMVERI) has organised an colloquium on “Groups : From a geometric point of view VI”, Lecture series by Dr. Oorna Mitra, Indian Statistical Institute on 15 March 2024.

Title: Groups : From a geometric point of view VI

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.