Construction of Type Ii1 Factors with Prescribed Countable Fundamental Group

In the context of Free Probability Theory, we study two different constructions that provide new examples of factors of type II1 with prescribed 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 = Z ⋊ SL(2,Z)). For any countable subgroup S ⊂ R∗+, we show that there exists an action of G on L(F∞) such that 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 Γ ⊂ R∗+ the subgroup generated by the point spectrum of ψ. We show that the centralizer (L(G)∗B(H)) is a type II1 factor and its fundamental group is Γ. Our proofs rely on Popa’s deformation/rigidity strategy using his intertwining-by-bimodules technique.