Octagonal relations

TL;DR

The paper investigates the action of the absolute Galois group on non-abelian fundamental groups arising from cyclic coverings of $\mathbb P^1$ punctured at $0$, 1, $\infty$ and roots of unity, introducing the octagonal relation as a central structural constraint. It builds a framework of $\,p$-adic measures $\beta_m(\sigma)$ encoding Galois action via $\ell$-adic polylogarithm data, derives the octagonal relation modulo augmentation ideal, and proves symmetry properties for $\beta_2(\sigma)$ that mirror Deligne-Drinfeld-Ihara relations. The work develops a formal calculus of measures, translations, and exterior products, connects to $p$-adic multiple zeta values through symmetrization (and proposes a route to $p$-adic multiple zeta functions), and provides corrections to earlier results while laying out the field-isomorphism machinery underpinning the approach. The results offer a rigorous path from Galois actions on fundamental groups to potential $p$-adic zeta-function-type invariants, with explicit structural identities and a solid foundation for further arithmetic applications.

Abstract

We study the action of the absolute Galois group on the fundamental groups.