Explicit examples of equivalence relations and factors with prescribed fundamental group and outer automorphism group

In this paper we give a number of explicit constructions for II$_1$ factors and II$_1$ equivalence relations that have prescribed fundamental group and outer automorphism group. We construct factors and relations that have uncountable fundamental group different from $\IRpos$. In fact, given any II$_1$ equivalence relation, we construct a II$_1$ factor with the same fundamental group. Given any locally compact unimodular second countable group $G$, our construction gives a II$_1$ equivalence relation $\RelR$ whose outer automorphism group is $G$. The same construction does not give a II$_1$ factor with $G$ as outer automorphism group, but when $G$ is a compact group or if $G=\SL^{\pm}_n\IR=\{g\in\GL_n\IR\mid \det(g)=\pm1\}$, then we still find a type II$_1$ factor whose outer automorphism group is $G$.