Amalgamated Free Products of $w$-Rigid Factors and Calculation of their Symmetry Groups

We consider amalgamated free product II${}_1$ factors $M=M_1{\ast }_B{M}_2{\ast }_B\mathrm{.}\mathrm{.}\mathrm{.}$ and use deformation/rigidity'' and intertwining’’ techniques to prove that any relatively rigid von Neumann subalgebra $Q\subset M$ can be intertwined into one of the $M_i$’s. We apply this to the case $M_i$ are w-rigid II${}_1$ factors, with $B$ equal to either $\mathbb{C}$, to a Cartan subalgebra $A$ in $M_i$, or to a regular hyperfinite II${}_1$ subfactor $R$ in $M_i$, to obtain the following type of unique decomposition results, `a la Bass-Serre: If $M = (N_1 *_C N_2 *_C …)^t$,forsome$t>0$ and some other similar inclusions of algebras $C\subset N_j$ then, after a permutation of indices, $(B\subset M_i)$ is inner conjugate to $(C\subset N_i{)}^t$, $\mathrm{\forall }i$. Taking $B=\mathbb{C}$ and $M_i=(L({\mathbb{Z}}^2⋊{\mathbb{F}}_2){)}^{t_i}$, with $\{t_i{\}}_{i\ge 1}=S$ a given countable subgroup of ${\mathbb{R}}_{+}^{\ast }$, we obtain continuously many non stably isomorphic factors $M$ with fundamental group $\text{mycal}F(M)$ equal to $S$. For $B=A$, we obtain a new class of factors $M$ with unique Cartan subalgebra decomposition, with a large subclass satisfying $\text{mycal}F(M)=\{1\}$ and Out$(M)$ abelian and calculable. Taking $B=R$, we get examples of factors with $\text{mycal}F(M)=\{1\}$, Out$(M)=K$, for any given separable compact abelian group $K$.