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Introduction to ℓ²-invariants

  • Holger Kammeyer
Book

Part of the Lecture Notes in Mathematics book series (LNM, volume 2247)

Table of contents

  1. Front Matter
    Pages i-viii
  2. Holger Kammeyer
    Pages 1-8
  3. Holger Kammeyer
    Pages 9-33
  4. Holger Kammeyer
    Pages 35-66
  5. Holger Kammeyer
    Pages 67-86
  6. Holger Kammeyer
    Pages 87-126
  7. Holger Kammeyer
    Pages 127-163
  8. Back Matter
    Pages 165-183

About this book

Introduction

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.

Keywords

ℓ ²-invariants ℓ ²-Betti Numbers ℓ ²-torsion Lück Approximation Torsion Growth

Authors and affiliations

  • Holger Kammeyer
    • 1
  1. 1.Institute for Algebra and GeometryKarlsruhe Institute of TechnologyKarlsruheGermany

Bibliographic information

  • DOI https://doi.org/10.1007/978-3-030-28297-4
  • Copyright Information Springer Nature Switzerland AG 2019
  • Publisher Name Springer, Cham
  • eBook Packages Mathematics and Statistics
  • Print ISBN 978-3-030-28296-7
  • Online ISBN 978-3-030-28297-4
  • Series Print ISSN 0075-8434
  • Series Online ISSN 1617-9692
  • Buy this book on publisher's site