On the fundamental group of II$_1$ factors and equivalence relations arising from group actions....

last revised 6 Oct 2008 (this version, v2)) Abstract: Given a countable group $G$, we consider the sets $\Sfactor(G)$, $\Sequiv(G)$, of subgroups $H \subset \R$ for which there exists a free ergodic probability measure preserving action $G \actson X$ such that the fundamental group of the associated II$_1$ factor $\rL^\infty(X)\rtimes G$, respectively orbit equivalence relation $\cR(G\actson X)$, equals $\exp(H)$. We prove that if $G=\Gamma^{*\infty}* \Bbb Z$, with $\Gamma\neq 1$, then $\Sfactor(G)$ and $\Sequiv(G)$... [read full story]