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

Deprez, Steven

doi:10.1090/tran/6298

Explicit examples of equivalence relations and II$_1$ factors with prescribed fundamental group and outer automorphism group
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Trans. Amer. Math. Soc. 367 (2015), 6837-6876 Request permission

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 $\mathbb {R}_{+}^{\ast }$. 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 $\mathcal {R}$ 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 $G=\mathrm {SL}^{\pm }_n\mathbb {R}=\{g\in \mathrm {GL}_n\mathbb {R}\mid \det (g)=\pm 1\}$, then we still find a type II$_1$ factor whose outer automorphism group is $G$.

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