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#### TBDmathscidoc:1701.331991

Acta Mathematica, 200, (1), 85-153, 2006.2
We consider amalgamated free product II_{1}factors$M$=$M$_{1*$B$}$M$_{2*$B$}… 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$M$_{$i$}’s. We apply this to the case where the$M$_{$i$}’s are w-rigid II_{1}factors, with$B$equal to either$C$, 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, àla Bass–Serre: If$M$= ($N$_{1 * C}N_{2*$C$}…)^{$t$}, for some$t$> 0 and some other similar inclusions of algebras$C$⊂$N$_{$i$}then, after a permutation of indices, ($B$⊂$M$_{$i$}) is inner conjugate to ($C$⊂$N$_{$i$})^{$t$}, for all$i$. Taking$B$=$C$and $M_{i} = {\left( {L{\left( {Z^{2} \rtimes F_{2} } \right)}} \right)}^{{t_{i} }}$ , with {$t$_{$i$}}_{$i$⩾1}=$S$a given countable subgroup of$R$_{+}^{*}, we obtain continuously many non-stably isomorphic factors$M$with fundamental group ${\user1{\mathcal{F}}}{\left( M \right)}$ equal to$S$. For$B$=$A$, we obtain a new class of factors$M$with unique Cartan subalgebra decomposition, with a large subclass satisfying ${\user1{\mathcal{F}}}{\left( M \right)} = {\left\{ 1 \right\}}$ and Out(M) abelian and calculable. Taking$B$=$R$, we get examples of factors with ${\user1{\mathcal{F}}}{\left( M \right)} = {\left\{ 1 \right\}}$ , Out($M$) =$K$, for any given separable compact abelian group$K$.
@inproceedings{adrian2006amalgamated,
title={Amalgamated free products of weakly rigid factors and calculation of their symmetry groups},
author={Adrian Ioana, Jesse Peterson, and Sorin Popa},
url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170108203351583487700},
booktitle={Acta Mathematica},
volume={200},
number={1},
pages={85-153},
year={2006},
}

Adrian Ioana, Jesse Peterson, and Sorin Popa. Amalgamated free products of weakly rigid factors and calculation of their symmetry groups. 2006. Vol. 200. In Acta Mathematica. pp.85-153. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170108203351583487700.