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November 2011 On the combinatorial anabelian geometry of nodally nondegenerate outer representations
Yuichiro Hoshi, Shinichi Mochizuki
Hiroshima Math. J. 41(3): 275-342 (November 2011). 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


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Yuichiro Hoshi. Shinichi Mochizuki. "On the combinatorial anabelian geometry of nodally nondegenerate outer representations." Hiroshima Math. J. 41 (3) 275 - 342, November 2011.


Published: November 2011
First available in Project Euclid: 12 December 2011

zbMATH: 1264.14041
MathSciNet: MR2895284
Digital Object Identifier: 10.32917/hmj/1323700038

Primary: 14H30
Secondary: 14H10

Keywords: combinatorial anabelian geometry , combinatorial cuspidalization , hyperbolic curve , Injectivity , nodally nondegenerate , outer Galois representation , semi-graph of anabelioids

Rights: Copyright © 2011 Hiroshima University, Mathematics Program

Vol.41 • No. 3 • November 2011
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