Introduction to ?²-invariants

by Holger Kammeyer

Overview

This book introduces the reader to the most important concepts and problems in the field of ℓ²-invariants. After some foundational material on group von Neumann algebras, ℓ²-Betti numbers are defined and their use is illustrated by several examples. The text continues with Atiyah's question on possible values of ℓ²-Betti numbers and the relation to Kaplansky's zero divisor conjecture. The general definition of ℓ²-Betti numbers allows for applications in group theory. A whole chapter is dedicated to Lück's approximation theorem and its generalizations. The final chapter deals with ℓ²-torsion, twisted variants and the conjectures relating them to torsion growth in homology. The text provides a self-contained treatment that constructs the required specialized concepts from scratch. It comes with numerous exercises and examples, so that both graduate students and researchers will find it useful for self-study or as a basis for an advanced lecture course.

Product Details

ISBN-13:	9783030282974
Publisher:	Springer-Verlag New York, LLC
Publication date:	10/29/2019
Series:	Lecture Notes in Mathematics , #2247
Sold by:	Barnes & Noble
Format:	eBook
File size:	10 MB

About the Author

Holger Kammeyer studied Mathematics at Göttingen and Berkeley. After a postdoc position in Bonn he is now based at Karlsruhe Institute of Technology. His research interests range around algebraic topology and group theory. The application of ℓ ²-invariants forms a recurrent theme in his work. He has given introductory courses on the matter on various occasions.

Table of Contents

- Introduction. - Hilbert Modules and von Neumann Dimension. - l2-Betti Numbers of CW Complexes. - l2-Betti Numbers of Groups. - Lück’s Approximation Theorem. - Torsion Invariants.

Editorial Reviews

This is an excellent introductory book, to be recommended to readers looking for an introduction to the field, as well as those that want to have an overview of recent developments.” (Joan Porti, Mathematical Reviews, September, 2020)

From the Publisher