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. We then define and study the category of homogeneous quasigroups, whose objects are groups and whose morphisms are equivalence classes of quasimorphisms in our sense. We call the automorphism group QOut(G) of a group G in this category the quasioutomorphism group. It acts on the space of real-valued homogeneous quasimorphisms on G extending the natural action of Out(G). We discuss various classes of examples of quasioutomorphisms of free groups. We use these examples to show that the orbit of Hom(F_n, R) under QOut(F_n) spans a dense subspace. This is contrast to the classical fact that the corresponding Out(F_n)-orbit is closed and of uncountable codimension. We also show that QOut(Z^n) = GL_n(R).