Abstract
The theory of von Neumann algebras can be viewed as a model of non-commutative measure theory. Therefore, it is plausible that L2-Betti numbers can be computed in terms of probability measure preserving actions.
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On the one hand, this leads to an additional way of computing L2-Betti numbers of groups.
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On the other hand, in this way, L2-Betti numbers provide orbit equivalence invariants.
We will first recall basic terminology from measured group theory. Then we will study L2-Betti numbers of standard equivalence relations. Moreover, we will discuss cost and its relation with the first L2-Betti number and rank gradients.
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Löh, C. (2020). The Dynamical View: Measured Group Theory. In: Ergodic Theoretic Methods in Group Homology. SpringerBriefs in Mathematics. Springer, Cham. https://doi.org/10.1007/978-3-030-44220-0_5
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DOI: https://doi.org/10.1007/978-3-030-44220-0_5
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Publisher Name: Springer, Cham
Print ISBN: 978-3-030-44219-4
Online ISBN: 978-3-030-44220-0
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