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

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

Hom-versions of the Combinatorial Grothendieck Conjecture II: Outer Representations of PIPSC- and NN-type

In the present paper, we continue our study, which was initiated in the previous paper of the present series of papers, of combinatorial anabelian geometry of (not necessarily bijective) continuous homomorphisms between PSC-fundamental groups of semi-graphs of anabelioids of PSC-type. In particular, we continue to study certain Hom-versions of the combinatorial versions of the Grothendieck conjecture established in some previous works, i.e., to study certain sufficient conditions of certain group-theoretic compatibility properties described in terms of outer representations. The outer representations we mainly concern in the present paper are of PIPSC-type and of NN-type, both of which are of substantial importance in the study of algebro-geometric anabelian geometry of configuration spaces of hyperbolic curves. We also include, as a preparation for one of the main results, a presentation of a “reduction technique”, namely, a technique of reduction to the “compactified quotients” of (various open subgroups of) the PSC-fundamental groups under consideration, in a similar vein to the previous paper where we included other two “reduction techniques”

Reconstruction of inertia groups associated to log divisors from a configuration space group equipped with its collection of log-full subgroups

In the present paper, we study configuration space groups. The goal of this paper is to reconstruct group-theoretically the inertia groups associated to various types of log divisors of a log configuration space of a smooth log curve from the associated configuration space group equipped with its collection of log-full subgroups

Good reduction of hyperbolic polycurves and their fundamental groups : A survey (Algebraic Number Theory and Related Topics 2018)

Algebraic Number Theory and Related Topics 2018. November 26-30, 2018. edited by Takao Yamazaki and Shuji Yamamoto. The papers presented in this volume of RIMS Kôkyûroku Bessatsu are in final form and refereed.The goal of this manuscript is to provide a survey of good reduction criteria for hyperbolic polycurves. In particular, we give outlines of the proofs of the main theorems of the papers [19] and [20], which are details of the talk “Criteria for good reduction of hyperbolic polycurves” given at “Algebraic Number Theory and Related Topics 2018”. Also, this paper contains a proof of a specialization theorem of pro-L fundamental groups

Tripod-degrees

Let p, l be distinct prime numbers. A tripod-degree over p at l is defined to be an l-adic unit obtained by forming the image, by the l-adic cyclotomic character, of some continuous automorphism of the geometrically pro-l fundamental group of a split tripod over a finite field of characteristic p. The notion of a tripod-degree plays an important role in the study of the geometrically pro-l anabelian geometry of hyperbolic curves over finite fields, e.g., in the theory of cuspidalizations of the geometrically pro-l fundamental groups of hyperbolic curves over finite fields. In the present paper, we study the tripod-degrees. In particular, we prove that, under a certain condition, the group of tripod-degrees over p at l coincides with the closed subgroup of the group of l-adic units topologically generated by p. As an application of this result, we also conclude that, under a certain condition, the natural homomorphism from the group of automorphisms of the split tripod to the group of outer continuous automorphisms of the geometrically pro-l fundamental group of the split tripod that lie over the identity automorphism of the absolute Galois group of the basefield is surjective

On the combinatorial cuspidalization of hyperbolic curves

In this paper, we continue our study of the pro-Σ fundamental groups of configuration spaces associated to a hyperbolic curve, where Σ is either the set of all prime numbers or a set consisting of a single prime number, begun in an earlier paper. Our main result may be regarded either as a combinatorial, partially bijective generalization of an injectivity theorem due to Matsumoto or as a generalization to arbitrary hyperbolic curves of injectivity and bijectivity results for genus zero curves due to Nakamura and Harbater-Schneps. More precisely, we show that if one restricts one’s attention to outer automorphisms of such a pro-Σ fundamental group of the configuration space associated to a(n) affine (respectively, proper) hyperbolic curve which are compatible with certain “fiber subgroups ” [i.e., groups that arise as kernels of the various natural projections of a configuration space to lower-dimensional configuration spaces] as well as with certain cuspidal inertia subgroups, then, as one lowers the dimension of the configuration space under consideration from n +1 to n ≥ 1 (respectively, n ≥ 2), there is a natural injection between the resulting groups of such outer automorphisms, which is a bijection if n ≥ 4. The key tool in the proo

On Generalizations of Anabelian Group-theoretic Properties

In the present paper, we discuss certain generalizations on two anabelian group-theoretic properties --strong internal indecomposability and elasticity. More concretely, by replacing the normality conditions appearing in characterizations of strong internal indecomposability and elasticity by the subnormality conditions, we introduce the notions of strong sn-internal indecomposability and sn-elasticity and prove that various profinite groups appearing in anabelian geometry satisfy these properties

On pro-p anabelian geometry for hyperbolic curves of genus 0 over p-adic fields

Let p be a prime number. In the present paper, we discuss the relative／absolute version of the geometrically pro-p anabelian Grothendieck Conjecture (RpGC／ApGC). In the relative setting, we prove RpGC for hyperbolic curves of genus 0 over subfields of mixed characteristic valuation fields of rank 1 of residue characteristic p whose value groups have no nontrivial p-divisible element. In particular, one may take the completion of arbitrary tame extension of a mixed characteristic Henselian discrete valuation field of residue characteristic p as a base field. In light of the condition on base fields, this result may be regarded as a partial generalization of S. Mochizuki's classical anabelian result, i.e., RpGC for arbitrary hyperbolic curves over subfields of finitely generated fields of the completion of the maximal unramified extension of ℚp. It appears to the author that this result suggests that much wider class of p-adic fields may be considered as base fields in anabelian geometry. In the absolute setting, under the preservation of decomposition subgroups, we prove ApGC for hyperbolic curves of genus 0 over mixed characteristic Henselian discrete valuation fields of residue characteristic p. This result may be regarded as the first absolute Grothendieck Conjecture-type result for hyperbolic curves in the pro-p setting. Moreover, by combining this ApGC-type result with combinatorial anabelian geometry, under certain assumptions on decomposition groups and dimensions, we prove ApGC for configuration spaces of arbitrary hyperbolic curves over unramified extensions of p-adic local fields or their completions. In light of the condition on the dimension of configuration spaces, this result may be regarded as a partial generalization of a K. Higashiyama's pro-p semi-absolute Grothendieck Conjecture-type result