Hiroshima Mathematical Journal

On the combinatorial anabelian geometry of nodally nondegenerate outer representations

Yuichiro Hoshi and Shinichi Mochizuki

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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

Article information

Hiroshima Math. J., Volume 41, Number 3 (2011), 275-342.

First available in Project Euclid: 12 December 2011

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 14H30: Coverings, fundamental group [See also 14E20, 14F35]
Secondary: 14H10: Families, moduli (algebraic)

Hyperbolic curve outer Galois representation injectivity semi-graph of anabelioids nodally nondegenerate combinatorial anabelian geometry combinatorial cuspidalization


Hoshi, Yuichiro; Mochizuki, Shinichi. On the combinatorial anabelian geometry of nodally nondegenerate outer representations. Hiroshima Math. J. 41 (2011), no. 3, 275--342. doi:10.32917/hmj/1323700038. https://projecteuclid.org/euclid.hmj/1323700038

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