Unambiguous injective morphisms in free groups

A morphism g is ambiguous with respect to a word u if there exists a second morphism h≠g such that g(u)=h(u). Otherwise g is unambiguous with respect to u. Thus unambiguous morphisms are those for which the structure of the morphism is preserved in the image. Ambiguity has so far been studied for morphisms of free monoids, where several characterisations exist for the set of words u permitting an (injective) unambiguous morphism. In the present paper, we consider ambiguity of morphisms of free groups, and consider possible analogies to the existing characterisations in the free monoid. While a direct generalisation results in a trivial situation where all morphisms are ambiguous, we discuss some natural and well-motivated reformulations, and provide a characterisation of words in a free group that permit a morphism which is “as unambiguous as possible”.