Given II_1 factors N and M, we will explain how to construct a group out of Hom(N,M).  It carries natural actions of the outer automorphism groups of N and M, hence there are cases where the fundamental group of M acts on it too.  In fact, in many interesting examples the action of the fundamental group of M gives rises to a natural vector space structure and then the group we construct turns out to be a Banach space.  This is joint work with Valerio Capraro.