# Comments on "Discrete Groups, Expanding Graphs and Invariant Measures", by Alexander Lubotzky

## TL;DR

This document offers detailed comments, clarifications, and proofs on the first four chapters of Lubotzky's book, aiming to complement and deepen understanding of discrete groups, expanding graphs, and invariant measures.

## Contribution

It provides detailed annotations, corrections, and supplementary proofs to enhance comprehension of Lubotzky's foundational work in the specified mathematical areas.

## Key findings

- Clarifications of proofs in the first four chapters
- Identification of imprecisions and their discussions
- Additional proofs of referenced facts

## Abstract

This document is a collection of comments that I wrote down while reading the first four chapters of the book "Discrete Groups, Expanding Graphs and Invariant Measures" by Alexander Lubotzky. Most of them are more detailed versions of proofs. Some imprecisions are pointed out and discussed, and some facts referenced in the book are proven. In the appendix we discuss topics of interest in relation to this book, which are however not necessary for its understanding. The aim of this document, which is not quite complete in that respect, is to provide, together with Lubotzky's book, a self-contained read.