On the combinatorial anabelian geometry of nodally nondegenerate outer representations

• Published in: Hiroshima mathematical journal 2011-11, Vol.41 (3), p.275-342
• Main Authors: Hoshi, Yuichiro, Mochizuki, Shinichi
• Format: Article
• Language: English
• Subjects: 14H10; 14H30; combinatorial anabelian geometry; combinatorial cuspidalization; Hyperbolic curve; injectivity; nodally nondegenerate; outer Galois representation; semi-graph of anabelioids
• ISSN: 0018-2079
• DOI: 10.32917/hmj/1323700038

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 Matsumoto to the case of proper hyperbolic curves