# Growth and expansion in algebraic groups over finite fields

## TL;DR

This paper introduces the study of growth in algebraic groups over finite fields, focusing on groups like SL_2(F_q) and their subgroups, highlighting recent developments and key examples in the area.

## Contribution

It provides an accessible overview of growth phenomena in groups of Lie type, synthesizing recent research and course notes for educational purposes.

## Key findings

- Analysis of growth in SL_2(F_q) and subgroups
- Connections to algebraic group theory over finite fields
- Summary of recent advances in the field

## Abstract

This is a brief introduction to the study of growth in groups of Lie type, with $SL_2(\mathbb{F}_q)$ and some of its subgroups as the key examples. They are an edited version of the notes I distributed at the Arizona Winter School in 2016. Those notes were, in turn, based in part on my survey in Bull. Am. Math. Soc. (2015) and in part on the notes for courses I gave on the subject in Cusco (AGRA) and G\"ottingen.