Unambiguous injective morphisms in free groups

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