Combinatorial Belyi Cuspidalization and Arithmetic Subquotients of the Grothendieck–Teichmüller Group

Abstract

In this paper, we develop a certain combinatorial version of the theory of Belyi cuspidalization developed by Mochizuki. Write $\overline{\mathbb{Q}}\subseteq \mathbb{C}$ for the subfield of algebraic numbers $\in \mathbb{C}$. We then apply this theory of combinatorial Belyi cuspidalization to certain natural closed subgroups of the Grothendieck–Teichmüller group associated to the field of $p$-adic numbers [where $p$ is a prime number] and to stably $\times mu$-indivisible subfields of $\overline{\mathbb{Q}}$, i.e., subfields for which every finite field extension satisfies the property that every nonzero divisible element in the field extension is a root of unity.