Hom-versions of the Combinatorial Grothendieck Conjecture I: Abelianizations and Graphically Full Actions

Abstract

Semi-graphs of anabelioids of PSC-type and their PSC-fundamental groups (i.e., a combinatorial Galois-category-theoretic abstraction of pointed stable curves over algebraically closed fields of characteristic zero and their fundamental groups) are central objects in the study of combinatorial anabelian geometry. In the present series of papers, which consists of two successive works, we investigate combinatorial anabelian geometry of (not necessarily bijective) continuous homomorphisms between PSC-fundamental groups. This contrasts with previous researches, which focused only on continuous isomorphisms. More specifically, our main results of the present series of papers roughly state that, if a continuous homomorphism between PSC-fundamental groups is compatible with certain outer representations, then it satisfies a certain “group-theoretic compatibility property”, i.e., the property that each of the images via the continuous homomorphism of certain VCN-subgroups of the domain are included in certain VCNsubgroups of the codomain. Such results may be considered as Homversions of the combinatorial version of the Grothendieck conjecture established in some previous works. As in the case of previous works (i.e., the Isom-versions), the proof requires different techniques depending on the types of outer representations under consideration. In the present paper, we will treat the case where the outer representations under consideration are assumed to be “l-graphically full”, i.e., to satisfy a certain condition concerning “weights” considered with respect to the “l-adic cyclotomic character”, where l is a certain prime number. In addition, to prepare for this purpose, we include detailed expositions on “reduction techniques”, namely, techniques of reduction to the maximal pro-Σ quotients and to the abelianizations of (various open subgroups of) the PSC-fundamental groups under consideration, where Σ is a certain set of prime numbers. Though the discussions of these “reduction techniques” are all essentially wellknown to experts, we present the results in a highly unified／generalized fashion.