On quasioutomorphism groups of free groups and their transitivity properties

We introduce a notion of quasimorphism between two arbitrary groups, generalizing the classical notion of Ulam. The collection of invertible equivalence classes of quasimorphisms from a group G   to itself forms a group QOut(G)QOut(G), which we call the quasioutomorphism group of G, since its action on the space of real-valued homogeneous quasimorphisms on G   extends the natural Out(G)Out(G)-action. We show that QOut(G)=GLn(R)QOut(G)=GLn(R) for every finitely generated amenable group G  . We then study quasioutomorphism groups of finitely generated non-abelian free groups and show that the orbit of Hom(Fn,R)Hom(Fn,R) under QOut(Fn)QOut(Fn) spans a dense subspace in the space of homogeneous quasimorphisms on FnFn. This is in contrast to the classical fact that the corresponding Out(Fn)Out(Fn)-orbit is closed and of uncountable codimension.