UDGRP 2024
Welcome to the Undergraduate Directed Group Reading Program (UDGRP) 2024! This year, we will explore Geometric Group Theory (GGT).
In the initial lectures, we’ll review basic group theory concepts and provide problem sets periodically. Please make a sincere effort to solve them, as this practice will prove invaluable in the long run. If you have any doubts, feel free to reach out to any of the instructors—we’re here to help!
To make things more engaging, we’ll introduce a “Chocolate Problem” at the end of each class. Anyone who provides a meaningful attempt—be it partial, intuitive, or without a rigorous proof—will be rewarded with chocolates! (Sadly, we can’t hand out Fields Medals, but hey, chocolates are a good start, right?)
Our goal with this approach is to encourage you to engage with the material actively. Mathematics is a discipline where simply knowing the theory isn’t enough—you must learn how to apply it effectively. And the best way to do so? Solve as many problems as possible!
You’ll find all the lecture notes and problem sets on this website. If you miss a lecture, don’t worry—we’ll keep everything updated here so you can catch up.
Timeline
0. Introductory Talk
Date: October 19, 2024.
Speakers: Rinkiny Ghatak and Treanungkur Mal.
Rinkiny began by briefly introducing the definition of a group and providing some examples. She then defined free groups and explored the structure of Cayley graphs, including how the Cayley graph of a free group looks. Additionally, she introduced the concept of group presentations, concluding her talk with the group presentation of the well-known Lamplighter Group.
Whereas I have tried to demonstrate how one can understand a complex group presentation, such as that of the Lamplighter Group, by using the analogy of street lamps. Interestingly, the seemingly intricate presentation simplifies to switching finitely many lamps on an infinite street comprising infinitely many lamps. I also provided a brief definition of a group action, as this concept will be central to our study moving forward.
1. Basics of Group Theory
Date: November 25, 2024.
Speaker: Rinkiny Ghatak.
In this lecture, Rinkiny covered the basics of group theory, including:
- Definition of Group
- Examples of Group $\left( \mathbb{Z}, \mathbb{Z}/n\mathbb{Z}, \text{GL}_n(\mathbb{R}), \text{SL}_n(\mathbb{R}), S_n \right)$
- Subgroups
- Group Presentations (Generators and Relators)
- Structure preserving maps in Groups (Homomorphism and Isomorphism)
- Normal Subgroups and Cosets
- Kernal and Image of an Homomorphism
- The 1st Isomorphism Theorem
Lecture Notes: Download
Introductory Slides: Download
Problem Set: Download
Chocolate Problem’s Solution (by Nikhil Nagaria): Download
Correct Submissions to the Chocolate Problem (Lexicographical Order): Arkaprovo Das, Daibik Barik, Nikhil Nagaria, Payal Rajora, Ramdas Singh, Sai Prabhav, Sarvesh Soni, Shankha Suvra Dam.
2. Group Actions and Their Applications
Date: November 29, 2024.
Speaker: Treanungkur Mal.
In this lecture, I have covered the idea of group action and solved some problems using group action, including:
- Recap of Previous Class
- Idea of Quotienting Groups
- Applications of the 1st Isomorphism Theorem
- Motivation for Group Action
- Definition of Group Action
- Some Valid Group Actions
- Definition of Orbits and Stabilizer
- The Orbit Stabilizer Theorem
- Cayley’s Theorem (Only Statement)
- Some examples of group action $\left( \mathrm{SO}(2, \mathbb{R}) \text{ acts on } \mathbb{R}^2 \right)$
Lecture Notes: Download
Problem Set: Download
Chocolate Problem’s Solution (by Sai Prabhav): Download
Correct Submissions to the Chocolate Problem (Lexicographical Order): Arkaprovo Das, Daibik Barik, Nikhil Nagaria, Sai Prabhav, Sarvesh Soni.
3. Introduction to GGT
Date: December 6, 2024.
Speaker: Treanungkur Mal.
In this lecture, I covered the following topics and proved some theorems including:
- Basic notions of graph theory
- Generating sets of a group and group presentations
- Cayley graphs with respect to generating sets
- Definition of Free groups
- Group Action on Trees $\left(\mathbb{F}_2 \text{ acts on } \Gamma(G, S) \right)$
- Review of group actions and the induced homomorphism
- Types of group actions, focusing on isometric actions on metric spaces
- Isometric and free group actions on $\mathbb{R}^n$ $\implies$ torsion-free
- Brief Idea for the proof of Nielsen–Schreier Theorem
Lecture Notes: Download
Problem Set: Download
4. Quasi-Isometry in GGT
Date: December 12, 2024.
Speaker: Rinkiny Ghatak.
In this lecture, Rinkiny covered the following topics:
- Recalled the idea of the proof of Nielsen–Schreier Theorem
- Basic notions of metric spaces
- Path Metric on Groups
- Motivation and Definition of Quasi-Isometry
- Some Basic Examples and Proposition related to Quasi-Isometry
- Proof of $\mathbb{R}$ and $\mathbb{Z}$ being quasi isometric
Lecture Notes: Download
Problem Set: Download