  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.
