Construction of type $\mathrm{I}{\mathrm{I}}_1$ 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 $\mathrm{I}{\mathrm{I}}_1$ 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. $G={\mathbb{Z}}^2⋊\text{SL}(2,\mathbb{Z})$). For any countable subgroup $S\subset {\mathbb{R}}_{+}^{\ast }$, we show that there exists an action of $G$ on $L({\text{F}}_{\mathrm{\infty }})$ such that $L({\text{F}}_{\mathrm{\infty }})⋊G$ is a type $\mathrm{I}{\mathrm{I}}_1$ factor and its fundamental group is $S$. The second construction is based on a free product. Take $(B(H),\psi )$ any factor of type $\mathrm{I}$ endowed with a faithful normal state and denote by $\mathrm{\Gamma }\subset {\mathbb{R}}_{+}^{\ast }$ 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 $\mathrm{I}{\mathrm{I}}_1$ factor and its fundamental group is $\mathrm{\Gamma }$. Our proofs rely on Popa’s deformation/rigidity strategy using his intertwining-by-bimodules technique.