$L^2$-Betti numbers have played and continue to play an important role in group theory, often as obstructions to some properties of groups. If all $L^2$-Betti numbers vanish, then one can define a secondary invariant, the $L^2$-torsion, which is the $L^2$-analogue of the classical notion of Reidemeister torsion. It is a more sophisticated and richer but also harder to analyze invariant. In this talk we want to describe some of its applications and relations to group theory and 3-manifolds which should illustrate its potential. Finally we discuss some interesting open problems about $L^2$-torsion. The talk is meant as a survey and not as a talk conveying technical details.
Links:
[1] https://www.mpim-bonn.mpg.de/taxonomy/term/39
[2] https://www.mpim-bonn.mpg.de/node/3444
[3] https://www.mpim-bonn.mpg.de/HOGRO2024