Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

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
DOI: https://doi.org/10.1090/tran/6298
Published electronically: June 18, 2015
MathSciNet review: 3378816
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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 $G$.


References [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 46L36, 28D15, 46L40, 37A20

Retrieve articles in all journals with MSC (2010): 46L36, 28D15, 46L40, 37A20


Additional Information

Steven Deprez
Affiliation: Department of Mathematics, Katholieke Universiteit Leuven, Celestijnenlaan 200B, B-3001 Leuven, Belgium
Email: steven.f.l.deprez@gmail.com

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
Article copyright: © Copyright 2015 American Mathematical Society