Introduction to ℓ²-invariants

Beschreibung

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.
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.
An up-to-date and user-friendly introduction to the rapidly developing field of ℓ²-invariants Proceeds quickly to the research level after thoroughly covering all the basics Contains many motivating examples, illustrations, and exercises

Autor*in

Holger Kammeyer

Themen in »Introduction to ℓ²-invariants«

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

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“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)

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