Introduction to ℓ²-invariants, 1st ed. 2019 Lecture Notes in Mathematics Series, Vol. 2247

This book introduces the reader to the most important concepts and problems in the field of ?²-invariants. After some foundational material on group von Neumann algebras, ?²-Betti numbers are defined and their use is illustrated by several examples. The text continues with Atiyah's question on possible values of ?²-Betti numbers and the relation to Kaplansky's zero divisor conjecture. The general definition of ?²-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 ?²-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.

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