The growth of Betti numbers and approximation theorems

TL;DR

This paper introduces homology growth, proves Lück's approximation theorem, explores Betti number growth, and discusses related open problems and special cases like p-adic towers.

Contribution

It provides a proof of Lück's approximation theorem, discusses its generalizations, and examines Betti number growth in p-adic analytic towers.

Findings

Proof of Lück's approximation theorem

Discussion of Betti number growth and open problems

Analysis of p-adic analytic towers and related approximation theorems

Abstract

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.