On the fundamental group of II_1 factors and equivalence relations arising from group actions

Details On the fundamental group of II_1 factors and equivalence relations arising from group actions On the fundamental group of II_1 factors and equivalence relations arising from group actions

2008-10-03

arxiv.org/abs/0810.0706v4

In Quanta of Maths, Proceedings of the Conference in honor of A.Connes' 60th birthday, Clay Mathematics Institute Proceedings

Mathematics

Operator Algebras

Final version. We incorporated a result by Ershov: as a consequence, it is shown that free ergodic actions of non-icc property

Scientific paper

Given a countable group G, we consider the sets S_factor(G), S_eqrel(G), of subgroups F of the positive real line for which there exists a free ergodic probability measure preserving action G on X such that the fundamental group of the associated II_1 factor, respectively orbit equivalence relation, equals F. We prove that if G is the free product of Z and infinitely many copies of a non-trivial group \Gamma, then S_factor(G) and S_eqrel(G) contain R_+ itself, all of its countable subgroups, as well as uncountable subgroups whose log can have any Hausdorff dimension in the interval (0,1). We then prove that if G=\Gamma*\Lambda, with \Gamma, \Lambda finitely generated ICC groups, one of which has property (T), then S_factor(G)=S_eqrel(G)={1}. We also show that there exist II_1 factors M such that the fundamental group of M is R_+, but the associated II_\infty factor M tensor B(l^2) admits no continuous trace scaling action of R_+.

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