## Hiroshima Mathematical Journal

### On the combinatorial anabelian geometry of nodally nondegenerate outer representations

#### Abstract

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

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

Dates
First available in Project Euclid: 12 December 2011

https://projecteuclid.org/euclid.hmj/1323700038

Digital Object Identifier
doi:10.32917/hmj/1323700038

Mathematical Reviews number (MathSciNet)
MR2895284

Zentralblatt MATH identifier
1264.14041

Subjects
Secondary: 14H10: Families, moduli (algebraic)

#### Citation

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

#### References

• M. Boggi, The congruence subgroup property for the hyperelliptic modular group: the open surface case, Hiroshima Math. J. 39 (2009), 351–362.
• Y. Hoshi, Absolute anabelian cuspidalizations of configuration spaces over finite fields, Publ. Res. Inst. Math. Sci. 45 (2009), 661–744.
• E. Irmak, N. Ivanov, J. D. McCarthy, Automorphisms of surface braid groups, preprint (arXiv:math.GT/0306069v1 3 Jun 2003).
• F. Knudsen, The projectivity of the moduli space of stable curves. II. The stacks $\mcM_{g,n}$, Math. Scand. 52 (1983), 161–199.
• M. Matsumoto, Galois representations on profinite braid groups on curves, J. Reine Angew. Math. 474 (1996), 169–219.
• M. Matsumoto and A. Tamagawa, Mapping-class-group action versus Galois action on profinite fundamental groups, Amer. J. Math. 122 (2000), 1017–1026.
• S. Mochizuki, The Local Pro-$p$ Anabelian Geometry of Curves, Invent. Math. 138 (1999), 319–423.
• S. Mochizuki, The Absolute Anabelian Geometry of Hyperbolic Curves, Galois Theory and Modular Forms, Kluwer Academic Publishers (2003), 77–122.
• S. Mochizuki, Semi-graphs of anabelioids, Publ. Res. Inst. Math. Sci. 42 (2006), 221–322.
• S. Mochizuki, A combinatorial version of the Grothendieck conjecture, Tohoku Math J. 59 (2007), 455–479.
• S. Mochizuki, Absolute anabelian cuspidalizations of proper hyperbolic curves, J. Math. Kyoto Univ. 47 (2007), 451–539.
• S. Mochizuki, Topics in Absolute Anabelian Geometry II: Decomposition Groups and Endomorphisms, RIMS Preprint 1625 (2008); see http://www.kurims.kyoto-u.ac.jp/\verb| |motizuki/papers-english.html for a revised version.
• S. Mochizuki, On the Combinatorial Cuspidalization of Hyperbolic Curves, Osaka J. Math. 47 (2010), 651–715.
• S. Mochizuki and A. Tamagawa, The Algebraic and Anabelian Geometry of Configuration Spaces, Hokkaido Math. J. 37 (2008), 75–131.
• H. Nakamura, Galois rigidity of pure sphere braid groups and profinite calculus, J. Math. Sci. Univ. Tokyo, 1 (1994), 71–136.
• N. Takao, Braid monodromies on proper curves and pro-$l$ Galois representations, to appear in J. Inst. Math. Jussieu.