On the fundamental group of II_1 factors and equivalence relations arising from group actions

Abstract

Given a countable group G, we consider the sets S_factor(G), S_eqrel(G), of subgroups F of the positive real line for which there exists a free ergodic probability measure preserving action G on X such that the fundamental group of the associated II_1 factor, respectively orbit equivalence relation, equals F. We prove that if G is the free product of Z and infinitely many copies of a non-trivial group Γ, then S_factor(G) and S_eqrel(G) contain R_+ itself, all of its countable subgroups, as well as uncountable subgroups whose log can have any Hausdorff dimension in the interval (0,1). We then prove that if G=Γ*Λ, with Γ, Λfinitely generated ICC groups, one of which has property (T), then S_factor(G)=S_eqrel(G)={1}. We also show that there exist II_1 factors M such that the fundamental group of M is R_+, but the associated II_\infty factor M tensor B(l^2) admits no continuous trace scaling action of R_+.

Final version. We incorporated a result by Ershov: as a consequence, it is shown that free ergodic actions of non-icc property (T) groups can give rise to McDuff II_1 factors, in particular having fundamental group R_+

References in corpus (3)

• Orbit Equivalence and actions of F_n
• Cocycle and orbit superrigidity for lattices in SL(n,R) acting on homogeneous spaces
• Relative Property (T) Actions and Trivial Outer Automorphism Groups

Cited by in corpus (5)

• Unique Cartan decomposition for II_1 factors arising from arbitrary actions of hyperbolic groups
• Groups associated to $I{I}_1$-factors
• On fundamental groups of tensor product $\mathrm{I}{\mathrm{I}}_1$ factors
• Explicit examples of equivalence relations and factors with prescribed fundamental group and outer automorphism group
• Approximate equivalence of actions