In the Sakarovitch's book on automata theory, it is written in the introduction to the section on rationals in the free group that the material presented therein lays "the foundation of a truly mathematical theory of context-free languages". Nevertheless, this is not made explicit, as context-free languages and pushdown automata are beyond the scope of the book.
I am aware of some connections of free groups (and especially of what Sakarovitch calls involutive monoids) to the theory of pushdown automata and context-free languages -- e.g., the Dyck language, the Shamir's theorem, etc. However, I have had a hard time finding a source in which the "truly mathematical theory of context-free languages", mentioned by Sakarovitch, is actually built up.
The closest thing I have found is the Berstel's book on transductions and context-free languages. However, at first sight it seems to me that pushdown automata are treated only marginally in this book, while the theory of rational subsets of a free group is not applied at all. Perhaps the material I am looking for has been intended for Eilenberg's Volume C, but I am not sure about that neither.
So I would like to ask for a pointer to a book, survey, or perhaps a set of papers, from which I could learn something about the Sakarovitch's "truly mathematical theory of context-free languages" and its relations to free groups and their rational subsets. Or perhaps I am looking for something that does not actually exist?