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# Actions of F_\infty whose II_1 factors and orbit equivalence relations have prescribed fundamental group

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### Abstract

We show that given any subgroup F of R_+ which is either countable or belongs to a certain "large" class of uncountable subgroups, there exist continuously many free ergodic probability measure preserving actions \sigma_i of the free group with infinitely many generators such that their associated group measure space II_1 factors M_i and orbit equivalence relations R_i have fundamental group equal to F and with M_i (respectively R_i) stably non-isomorphic. Moreover, these actions can be taken so that R_i has no outer automorphisms and any automorphism of M_i is unitary conjugate to an automorphism that acts trivially on $$L^\infty(X_i) \subset M_i$$.

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###### Journal
2008-03-23
2008-06-02
10.1090/S0894-0347-09-00644-4
0803.3351

46L10, 28D15, 46L35, 20E05
Journal of the American Mathematical Society 23 (2010), 383--403
1) Minor changes w.r.t. v2. 2) Comment about v2: a modification of our main result provides now the first examples of separable II_1 factors and equivalence relations with uncountable fundamental group different from R_+
math.OA math.DS

Algebra