Composition of subfactors: New examples of infinite depth subfactors

Let N subset of P and P subset of M be inclusions of II1 factors with finite Jones index. We study the composed inclusion N subset of P subset of M by computing the fusion of N-P and P-M bimodules and determine various properties of N subset of M in terms of the ''small'' inclusions. A nice class of such subfactors arises in the following way: let H and K be two finite groups acting properly outerly on the hyperfinite II1 factor M and consider the inclusion M(H) subset of M x K. We show that properties like irreducibility, finite depth, amenability and strong amenability (in the sense of Popa) of M(H) subset of M x K can be expressed in terms of properties of the group G generated by H and K in OutM. In particular, the inclusion is amenable iff M is hyperfinite and the group G is amenable. We obtain many new examples of infinite depth subfactors (amenable and nonamenable ones), whose principal graphs have subexponential and/or exponential growth and can be determined explicitly. Furthermore, we construct irreducible, amenable subfactors of the hyperfinite II1 factor which are not strongly amenable.