  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_+.
