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

2010/10/18 by Deprez, Steven
#28D15 (Secondary) #46L36 (Primary) 37A20 #46L40 #Dynamical Systems (math.DS) #FOS: Mathematics #Group Theory (math.GR) #Operator Algebras (math.OA)

paper · doi:10.48550/arxiv.1010.3612

Abstract

In this paper we give a number of explicit constructions for II₁ factors and II₁ 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 II₁ equivalence relation, we construct a II₁ factor with the same fundamental group. Given any locally compact unimodular second countable group G, our construction gives a II₁ equivalence relation \RelR whose outer automorphism group is G. The same construction does not give a II₁ factor with G as outer automorphism group, but when G is a compact group or if G=\SL^±_n\IR=\g∈\GL_n\IR| det(g)=±1\, then we still find a type II₁ factor whose outer automorphism group is G.

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