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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Explicit examples of equivalence relations and II$_1$ factors with prescribed fundamental group and outer automorphism group
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by Steven Deprez PDF
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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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
  • © Copyright 2015 American Mathematical Society
  • 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
  • MathSciNet review: 3378816