Combinatorial Construction of the Absolute Galois Group of the Field of Rational Numbers

Abstract

In this paper, we give a purely combinatorial/group-theoretic construction of the conjugacy class of subgroups of the Grothendieck-Teichmüller group GT determined by the absolute Galois group GQ def = Gal(Q/Q) [where Q denotes the field of algebraic numbers] of the field of rational numbers Q. In fact, this construction also yields, as a by-product, a purely combinatorial/group-theoretic characterization of the GT-conjugates of closed subgroups of GQ that are "sufficiently large" in a certain sense. We then introduce the notions of TKND-fields [i.e., "torally Kummernondegenerate fields"] and AVKF-fields [i.e., "abelian variety Kummerfaithful fields"], which generalize, respectively, the notions of "torally Kummer-faithful fields" and "Kummer-faithful fields" [notions that appear in previous work of Mochizuki]. For instance, if we write Qab⊆ Q for the maximal abelian extension field of Q, then every finite extension of Qab is a TKND-AVKF-field [i.e., both TKND and AVKF]. We then apply the purely combinatorial/group-theoretic characterization referred to above to prove that, if a subfield K ⊆ Q is TKND-AVKF, then the commensurator in GT of the subgroup GK⊆ GQ determined by K is contained in GQ. Finally, we combine this computation of the commensurator with a result of Hoshi-Minamide-Mochizuki concerning GT to prove a semi-absolute version of the Grothendieck Conjecture for higher dimensional [i.e., of dimension ≥ 2] configuration spaces associated to hyperbolic curves of genus zero over TKND-AVKF-fields.