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

## TL;DR

This paper explores the Betti aspects of double shuffle theory, focusing on bitorsor structures and their relations to associators, completing a series of studies on the algebraic and geometric properties involved.

## Contribution

It explicitly constructs the Betti bitorsor structures and analyzes their discrete and pro-p versions within the double shuffle framework.

## Key findings

- Explicit description of Betti bitorsor structures
- Analysis of discrete and pro-p Betti groups
- Completion of the series' aim to connect Betti and de Rham aspects

## Abstract

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.