# Chip-Firing Games and Critical Groups

## TL;DR

This paper introduces the critical group of a finite connected graph, explaining its elementary combinatorial definition via chip-firing, and explores its properties and significance across various mathematical disciplines.

## Contribution

It provides an accessible exposition of the critical group, including basic definitions, properties, and open questions, aimed at undergraduate students.

## Key findings

- Defines the critical group using chip-firing operations
- Highlights connections to multiple areas of mathematics
- Includes exercises and open problems for further exploration

## Abstract

In this expository article intended to be accessible to undergraduate students we introduce a finite abelian group that can be associated to any finite connected graph. This group can be defined in an elementary combinatorial way in terms of chip-firing operations, and has been an object of interest in combinatorics, algebraic geometry, statistical physics, and several other areas of mathematics. We will begin with basic definitions and examples and develop a number of properties that can be derived by looking at this group from different angles. Throughout, we will give exercises, some of which are straightforward and some of which are open questions. We will also attempt to highlight some of the many contributions to this area made by undergraduate students