Explicit examples of equivalence relations and II₁ factors with prescribed fundamental group and outer automorphism group

Abstract

In this paper we give a number of explicit constructions for II 1 _1 factors and II 1 _1 equivalence relations that have prescribed fundamental group and outer automorphism group. We construct factors and relations that have uncountable fundamental group different from R + ∗ \mathbb {R}_{+}^{\ast } . In fact, given any II 1 _1 equivalence relation, we construct a II 1 _1 factor with the same fundamental group. Given any locally compact unimodular second countable group G G , our construction gives a II 1 _1 equivalence relation R \mathcal {R} whose outer automorphism group is G G . The same construction does not give a II 1 _1 factor with G G as outer automorphism group, but when G G is a compact group or if G = S L n ± R = { g ∈ G L n R ∣ det ( g ) = ± 1 } G=\mathrm {SL}^{\pm }_n\mathbb {R}=\{g\in \mathrm {GL}_n\mathbb {R}\mid \det (g)=\pm 1\} , then we still find a type II 1 _1 factor whose outer automorphism group is G G .