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

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

2010-10-18

arxiv.org/abs/1010.3612v2

Mathematics

Operator Algebras

version 2: added computations of outer automorphism groups

Scientific paper

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 $G$.

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