Subfactors of free products of rescalings of a II$_1$–factor
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Let Q be any II1-factor. It is shown that any standard lattice script G sign can be realized as the standard invariant of a free product of (several) rescalings of Q. In particular, if Q has fundamental group equal to the positive reals and if P is the free product of infinitely many copies of Q, then P has subfactors giving rise to all possible standard invariants. Similarly, given a II1-subfactor N ⊂ M, it is shown there are subfactors N̂ ⊂ M̂ having the same standard invariant as N ⊂ M but where M̂, respectively N̂, is the free product of M, respectively N, with rescalings of Q.