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

Abstract

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 \(G\).