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Semi-graphs of anabelioids. (English) Zbl 1113.14025
In a series of foregoing papers, the author has already very extensively pursued the general problem of developing an appropriate and efficient approach to representing, in the framework of abstract category theory, various scheme-theoretic geometries, among them being the absolute anabelian geometry of hyperbolic curves over $$p$$-adic local fields, the geometry of locally noetherian log schemes, the geometry of log schemes with archimedean structure, and others. Many of the author’s constructions, especially those concerning Grothendieck’s theory of the algebraic fundamental group of a Galois category as developed in SGA 1, turned out to be valid on a rather abstract category-theoretic level, at least so when working with the category of finite étale covers of a given base scheme.
In this context, the author invented his so-called “anabelioids”, which are basically Cartesian products of categories of finite sets equipped with a continuous action by a fixed profinite group $$G$$, and which serve to formalize the concept of a finite étale cover in a purely category-theoretic way the author [Publ. Res. Inst. Math. Sci. 40, No. 3, 819–881 (2004; Zbl 1113.14021)]. On the other hand, in his work on the anabelian geometry of hyperbolic curves over $$p$$-adic local fields, the author developed a certain geometry of “semi-graphs of profinite groups” [in: Galois theory and modular forms, Dev. Math. 11, 77–122 (2004; Zbl 1062.14031)]. Now, in the paper under review, these previously elaborated abstract theories are brought together in order to create an even more abstract, unifying framework, namely the formalism of “semi-graphs of anabelioids”. As the author points out, the aim of the present paper is to deliver “a piece of mathematical infrastructure” in maximal possible generality, that is, to exhibit both the formal basic properties of semi-graphs of anabelioids and the “general nonsense” they transpire, just to quote his own keywords. However, the author’s highly abstract approach leads to several interesting new results and deeper insights, including
(1) an analogue of Zariski’s main theorem for certain types of morphisms of semi-graphs;
(2) certain properties of the profinite fundamental group associated to a graph of anabelioids;
(3) a generalization of Y. André’s concept of tempered, fundamental groups [Duke Math. J. 119, No. 1, 1–39 (2003; Zbl 1155.11356)] to so-called “temperoids”;
(4) a localization theory for semi-graphs of anabelioids, with applications to the geometry of formal localizations of stable log curves;
(5) an analogue of the author’s anabelian geometry S. Mochizuki [Nagoya Math. J. 179, 17–45 (2005; Zbl 1129.14042)] with respect to tempered fundamental groups; and
(6) some new graph-theoretic applications to the study of free groups.
As for the presentation of all these abstract new concepts, methods, techniques, and results, the exhibition is fairly self-contained, lucid, detailed and well-structured. The author explains once again the basic notions from his foregoing papers as used in the present text, and he gives a plenty of motivations, supplementary remarks, hints for further reading, and concrete algebro-geometric examples.

##### MSC:
 14H30 Coverings of curves, fundamental group 14G32 Universal profinite groups (relationship to moduli spaces, projective and moduli towers, Galois theory) 14F35 Homotopy theory and fundamental groups in algebraic geometry 14H25 Arithmetic ground fields for curves 18E35 Localization of categories, calculus of fractions 20E05 Free nonabelian groups
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##### References:
 [1] André, Y., On a Geometric Description of Gal(Qp/Qp) and a p-adic Avatar of GT , Duke Math. J., 119 (2003), 1-39. · Zbl 1155.11356 [2] Deligne, P. and Mumford, D., The irreducibility of the moduli space of curves of given genus, IHES Publ. Math., 36 (1969), 75-109. · Zbl 0181.48803 [3] Dubuc, E., On the representation theory of Galois and atomic topoi, J. Pure Appl. Algebra, 186 (2004), 233-275. · Zbl 1037.18002 [4] Herfort, W. and Ribes, L., Torsion elements and centralizers in free products of profinite groups, J. Reine Angew. Math., 358 (1985), 155-161. · Zbl 0549.20017 [5] Knudsen, F. F., The projectivity of the moduli space of stable curves, II, Math. Scand., 52 (1983), 161-199. · Zbl 0544.14020 [6] Milne, J. S., Étale Cohomology, Princeton Mathematical Series, 33, Princeton University Press, 1980. · Zbl 0433.14012 [7] Mochizuki, S., The geometry of the compactification of the Hurwitz scheme, Publ. RIMS, Kyoto Univ., 31 (1995), 355-441. · Zbl 0866.14013
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