Anabelian perspectives in Galois-Teichm\"uller theory

Abstract

By exploiting the arithmetic homotopy of the moduli spaces of curves, Galois-Teichm\"uller theory stands at the interface of braid-mapping class groups and of anabelian geometry. Starting from the classical braid-theoretic construction of the Grothendieck-Teichm\"uller group, we review how anabelian geometry -- beginning with the foundational work of Nakamura -- provides the arithmetic mechanisms underlying its definition. We then explain how the combinatorial anabelian geometry developed by Hoshi and Mochizuki recasts these constructions within a purely group-theoretic and algorithmic framework. In particular, we describe how the group GT emerges as an anabelian object and how, once freed from auxiliary or artificially imposed containers, the anabelian algorithms yield a combinatorial reconstruction of the absolute Galois group of rational numbers. The perspective developed here highlights a conceptual shift from explicit braid-theoretic computations to functorial and algorithmic forms of anabelian reconstruction.