  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 $L^\infty(X_i) \subset M_i$.
