The growth of Betti numbers and approximation theorems

These short lecture notes provide a brief introduction to the field of homology growth. They are composed out of two lectures, which I have given at the Borel seminar 2017 in Les Diablerets. We give a proof of L"uck’s approximation theorem, discuss generalizations and mention some related open problems. Then we discuss the growth of mod-$p$ Betti numbers, where many problems remain open. We take a closer look at the special case of $p$-adic analytic towers and discuss an approximation theorem due to Bergeron-Linnell-L"uck-Sauer and Calegari-Emerton.