Construction of type ${\rm II_1}$ 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 ${\rm II_1}$ 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 $\mathcal{F}(L(G)) = \{1\}$ (e.g. $G = \Z^2 \rtimes \SL(2, \Z)$). For any countable subgroup $S \subset \R^*_+$, we show that there exists an action of $G$ on $L(\F_\infty)$ such that $L(\F_\infty) \rtimes G$ is a type ${\rm II_1}$ factor and its fundamental group is $S$. The second construction is based on a free product. Take $(B(H), \psi)$ any factor of type ${\rm I}$ endowed with a faithful normal state and denote by $\Gamma \subset \R^*_+$ the subgroup generated by the point spectrum of $\psi$. We show that the centralizer $(L(G) \ast B(H))^{\tau \ast \psi}$ is a type ${\rm II_1}$ factor and its fundamental group is $\Gamma$. Our proofs rely on Popa's deformation/rigidity strategy using his intertwining-by-bimodules technique.