Construction of type II₁ factors with prescribed countable fundamental group

Abstract

In the context of Free Probability Theory, we study two different constructions that provide new examples of factors of type II₁ with prescribed countable fundamental group. First we investigate state-preserving group actions on the almost periodic free Araki-Woods factors satisfying both a condition of mixing and a condition of free malleability in the sense of Popa. Typical examples are given by the free Bogoliubov shifts. Take an ICC w-rigid group G such that ℱ(L(G)) = {1} (e.g., G = ℤ² ⋊ SL(2, ℤ)). For any countable subgroup S ⊂ , we construct an action of G on L(𝔽_∞) such that the associated crossed product L(𝔽_∞) ⋊ G is a type II₁ factor and its fundamental group is S. The second construction is based on a free product. Take (B(H),ψ) any factor of type I endowed with a faithful normal state and denote by S ⊂ the subgroup generated by the point spectrum of ψ. We show that the centralizer (L(G) ∗ B(H))^τ*^ψ is a type II₁ factor and its fundamental group is S. Our proofs rely on Popa's deformation/rigidity strategy using his intertwining-by-bimodules technique.