On beta and gamma functions associated with the Grothendieck-Teichmüller group. Appendix: Profinite free differential calculus and profinite Blanchfield-Lyndon thereom (Q2712791)

Let \(\widehat{F}_2\) be the profinite completion of the free group on two generators, regarded as the algebraic fundamental group of \({\mathbb{P}}^1(\overline{\mathbb{Q}})\setminus\{0,1,\infty\}\) through the identification of generators with counter-clockwise loops around 0 and 1. There is an action of \(\text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})\) on \(\hat{F}_2\) which leads to \textit{G. W. Anderson}'s definition of hyperadelic gamma and beta functions, \(\Gamma_\sigma\) and \(B_\sigma\), \(\sigma \in \text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})\) [Invent. Math. 95, No. 1, 63--131 (1989; Zbl 0682.14011)]. Anderson proved a gamma factorization formula and a Gauss multiplication formula for these functions. In this paper the author generalizes the definition of these functions to \(\sigma\in \text{GT}\), the Grothendieck-Teichmüller group, which is a certain subgroup of \(\text{Aut}(\hat{F}_2)\) containing the image of \(\text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})\). He uses the 5-cycle relation satisfied by elements of \(\text{GT}\) to generalize the gamma factorization formula, and formulates a generalization of the Gauss multiplication formula. He suggests that this might be essentially arithmetic, and not always satisfied for \(\sigma \in \text{GT}\), and therefore could provide a way of distinguishing a proper subgroup of \(\text{GT}\) which contains the image of \(\text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})\).NEWLINENEWLINEFor part II of this paper see \textit{Y. Ihara}, J. Reine Angew. Math. 527, 1--11 (2000; see the preceding review Zbl 1046.14009).NEWLINENEWLINEFor the entire collection see [Zbl 0941.00014].