Fundamental groups of II_1 factors and equivalence relations

Other Titles:

Fundamentaalgroepen van II_1 factoren en equivalentie relaties

Authors:

Keersmaekers, Jan; S0165090

Issue Date:

11-Oct-2013

Abstract:

One invariant of II_1 factors and II_1 equivalence relations that has been studied extensively is the fundamental group. This notion is not related to the fundamental group of a topological space, but instead can be shown to be a subgroup of the positive real numbers. It was introduced by Murray and von Neumann for II1 factors. Murray and von Neumann were only able to compute it for the hyperfinite II_1 factor: F(R) = R*_+. Since then, much has been achieved, but it also became clear that calculating this fundamental group was very difficult. Whenever you have a countable group acting freely, ergodically and measure preservingly on a standard probability space, you can construct both a II_1 equivalence relation (the orbit equivalence relation), and a II1 factor (the group measure space construction). In this case, the fundamental group of the equivalence relation is a subgroup of the fundamental group of the II_1 factor. Many results involving these group actions show that equality holds in specific cases. However, this is not true in general. In 2006, Popa gave examples of group actions where the difference is as big as possible. In this thesis, I wil elaborate on this, giving examples where the equivalence relation can have arbitrary fundamental group, whereas the associated II1 factor has R*_+ as a fundamental group.Furthermore I give examples of another interesting phenomenon, where a II_1 factor contains a Cartan subalgebra such that the fundamental group of the equivalence relation associated to this Cartan subalgebra is non-trivial, whereas the fundamental group of the original II_1 factor is {1}. Indeed, whenever a II_1 factor has a Cartan subalgebra, Feldman and Moore showed that this gives riseto a II_1 equivalence relation and a scalar 2-cocycle. In our case, this 2-cocycle is non-trivial, and hence the equivalence relation associated to the Cartan subalgebra is twisted by a 2-cocycle inside the II_1 factor.