The Betti side of the double shuffle theory. III. Bitorsor structures

In the two first parts of the series, we constructed stabilizer subtorsors of a twisted Magnus' torsor, studied their relations with the associator and double shuffle torsors, and explained their de Rham’ nature. In this paper, we make the associated bitorsor structures explicit and explain the Betti' nature of the corresponding right torsors; we thereby complete one aim of the series. We study the discrete and pro-p versions of the Betti’ group of the double shuffle bitorsor.