paper

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

arXiv:1010.3612

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 $G$ .

version 2: added computations of outer automorphism groups