# Ergodic Theoretic Methods in Group Homology

## A Minicourse on L2-Betti Numbers in Group Theory

Book

Part of the SpringerBriefs in Mathematics book series (BRIEFSMATH)

1. Front Matter
Pages i-ix
2. Clara Löh
Pages 1-4
3. Clara Löh
Pages 5-15
4. Clara Löh
Pages 17-26
5. Clara Löh
Pages 27-35
6. Clara Löh
Pages 37-58
7. Clara Löh
Pages 59-71
8. Clara Löh
Pages 73-92
9. Back Matter
Pages 93-114

### Introduction

This book offers a concise introduction to ergodic methods in group homology, with a particular focus on the computation of L2-Betti numbers.

Group homology integrates group actions into homological structure. Coefficients based on probability measure preserving actions combine ergodic theory and homology. An example of such an interaction is provided by L2-Betti numbers: these invariants can be understood in terms of group homology with coefficients related to the group von Neumann algebra, via approximation by finite index subgroups, or via dynamical systems. In this way, L2-Betti numbers lead to orbit/measure equivalence invariants and measured group theory helps to compute L2-Betti numbers. Similar methods apply also to compute the rank gradient/cost of groups as well as the simplicial volume of manifolds.

This book introduces L2-Betti numbers of groups at an elementary level and then develops the ergodic point of view, emphasising the connection with approximation phenomena for homological gradient invariants of groups and spaces. The text is an extended version of the lecture notes for a minicourse at the MSRI summer graduate school “Random and arithmetic structures in topology” and thus accessible to the graduate or advanced undergraduate students. Many examples and exercises illustrate the material.

### Keywords

L2-Betti numbers von Neumann dimension measured group theory approximation of homological invariants simplicial volume ergodic theory of groups homological gradient invariants measured equivalence relations invariant random subgroups profinite completion

#### Authors and affiliations

1. 1.Fakultät für MathematikUniversität RegensburgRegensburgGermany

Clara Löh is Professor of Mathematics at the University of Regensburg, Germany. Her research focuses on simplicial volume and the interaction between geometric topology, geometric group theory, and measured group theory. This includes cohomological, geometric, and combinatorial methods.

### Bibliographic information

• Book Title Ergodic Theoretic Methods in Group Homology
• Book Subtitle A Minicourse on L2-Betti Numbers in Group Theory
• Authors Clara Löh
• Series Title SpringerBriefs in Mathematics
• Series Abbreviated Title SpringerBriefs in Mathematics
• DOI https://doi.org/10.1007/978-3-030-44220-0
• Copyright Information The Author(s), under exclusive license to Springer Nature Switzerland AG 2020
• Publisher Name Springer, Cham
• eBook Packages Mathematics and Statistics Mathematics and Statistics (R0)
• Softcover ISBN 978-3-030-44219-4
• eBook ISBN 978-3-030-44220-0
• Series ISSN 2191-8198
• Series E-ISSN 2191-8201
• Edition Number 1
• Number of Pages IX, 114
• Number of Illustrations 1 b/w illustrations, 0 illustrations in colour
• Topics
• Buy this book on publisher's site

## Reviews

“This is an attractive brisk introduction to the field, and will be a useful entry point to what is now a large and active field.” (Thomas B. Ward, zbMATH 1444.37001, 2020)