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On the combinatorial anabelian geometry of nodally nondegenerate outer representations. (English) Zbl 1264.14041
Summary: Let $$Sg$$ be a nonempty set of prime numbers. In the present paper, we continue the study, initiated in a previous paper by the second author, of the combinatorial anabelian geometry of semi-graphs of anabelioids of pro-$$Sg$$ PSC-type, i.e., roughly speaking, semi-graphs of anabelioids associated to pointed stable curves. Our first main result is a partial generalization of one of the main combinatorial anabelian results of this previous paper to the case of nodally nondegenerate outer representations, i.e., roughly speaking, a sort of abstract combinatorial group-theoretic generalization of the scheme-theoretic notion of a family of pointed stable curves over the spectrum of a discrete valuation ring. We then apply this result to obtain a generalization, to the case of proper hyperbolic curves, of a certain injectivity result, obtained in another paper by the second author, concerning outer automorphisms of the pro-$$Sg$$ fundamental group of a configuration space associated to a hyperbolic curve, as the dimension of this configuration space is lowered from two to one. This injectivity allows one to generalize a certain well-known injectivity theorem of M. Matsumoto [J. Reine Angew. Math. 474, 169–219 (1996; Zbl 0858.12002)] to the case of proper hyperbolic curves.

##### MSC:
 14H30 Coverings of curves, fundamental group 14H10 Families, moduli of curves (algebraic)
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