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Explicit examples of equivalence relations and II$ _1$ factors with prescribed fundamental group and outer automorphism group

Author: Steven Deprez
Journal: Trans. Amer. Math. Soc. 367 (2015), 6837-6876
MSC (2010): Primary 46L36; Secondary 28D15, 46L40, 37A20
Published electronically: June 18, 2015
MathSciNet review: 3378816
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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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Additional Information

Steven Deprez
Affiliation: Department of Mathematics, Katholieke Universiteit Leuven, Celestijnenlaan 200B, B-3001 Leuven, Belgium

Received by editor(s): October 5, 2012
Received by editor(s) in revised form: April 9, 2013
Published electronically: June 18, 2015
Additional Notes: The author was a research assistant of the Research Foundation – Flanders (FWO) (until August 2011) and a postdoc at the University of Copenhagen (from September 2011). The author was partially supported by ERC Grant VNALG-200749 and ERC Advanced Grant no. OAFPG 247321, and was supported by the Danish National Research Foundation (DNRF) through the Centre for Symmetry and Deformation
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