The growth of Betti numbers and approximation theorems

Steffen Kionke

The growth of Betti numbers and approximation theorems

Algebraic Topology 2017-09-05 v1 Geometric Topology

Authors: Steffen Kionke

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.

Keywords

homology growth Betti numbers L"uck's approximation theorem mod-$p$ Betti numbers $p$-adic analytic towers Bergeron-Linnell-L"uck-Sauer theorem Calegari-Emerton theorem

Cite

@article{arxiv.1709.00769,
  title  = {The growth of Betti numbers and approximation theorems},
  author = {Steffen Kionke},
  journal= {arXiv preprint arXiv:1709.00769},
  year   = {2017}
}

Comments

Borel seminar 2017 (14 pages)

The growth of Betti numbers and approximation theorems

Abstract

Keywords

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