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.