Amalgamated free products of weakly rigid factors and calculation of their symmetry groups

We consider amalgamated free product II1 factors M = M1*BM2*B… and use “deformation/rigidity” and “intertwining” techniques to prove that any relatively rigid von Neumann subalgebra Q ⊂ M can be unitarily conjugated into one of the Mi’s. We apply this to the case where the Mi’s are w-rigid II1 factors, with B equal to either C, to a Cartan subalgebra A in Mi, or to a regular hyperfinite II1 subfactor R in Mi, to obtain the following type of unique decomposition results, àla Bass–Serre: If M = (N1 * CN2*C…)t, for some t > 0 and some other similar inclusions of algebras C ⊂ Ni then, after a permutation of indices, (B ⊂ Mi) is inner conjugate to (C ⊂ Ni)t, for all i. Taking B = C and Mᵢ = ( L( Z² F₂ ) )^tᵢ, with {ti}i⩾1 = S a given countable subgroup of R+*, we obtain continuously many non-stably isomorphic factors M with fundamental group 1F( 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 1F( M ) = \ 1 \ and Out(M) abelian and calculable. Taking B = R, we get examples of factors with 1F( M ) = \ 1 \, Out(M) = K, for any given separable compact abelian group K.