The Dynamical View: Measured Group Theory
Part of the SpringerBriefs in Mathematics book series (BRIEFSMATH)
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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.
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.
On the one hand, this leads to an additional way of computing L2-Betti numbers of groups.
On the other hand, in this way, L2-Betti numbers provide orbit equivalence invariants.
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© The Author(s), under exclusive license to Springer Nature Switzerland AG 2020