The Arithmetic of Fundamental Groups pp 167-207 | Cite as

# On Monodromically Full Points of Configuration Spaces of Hyperbolic Curves

## Abstract

We introduce and discuss the notion of monodromically full points of configuration spaces of hyperbolic curves. This notion leads to complements to M. Matsumoto’s result concerning the difference between the kernels of the natural homomorphisms associated to a hyperbolic curve and its point from the Galois group to the automorphism and outer automorphism groups of the geometric fundamental group of the hyperbolic curve. More concretely, we prove that any hyperbolic curve over a number field has many nonexceptional closed points, i.e., closed points which do not satisfy a condition considered by Matsumoto, but that there exist infinitely many hyperbolic curves which admit many exceptional closed points, i.e., closed points which do satisfy the condition considered by Matsumoto. Moreover, we prove a Galois-theoretic characterization of equivalence classes of monodromically full points of configuration spaces, as well as a Galois-theoretic characterization of equivalence classes of quasi-monodromically full points of cores. In a similar vein, we also prove a necessary and sufficient condition for quasi-monodromically full Galois sections of hyperbolic curves to be geometric.

## Keywords

Exact Sequence Prime Number Fundamental Group Closed Point Galois Group## Preview

Unable to display preview. Download preview PDF.

## References

- AI88.G. Anderson and Y. Ihara. Pro-
*ℓ*branched coverings of*ℙ*^{1}and higher circular*ℓ*-units.*The Annals of Mathematics*, 128(2):271–293, 1988.CrossRefMATHMathSciNetGoogle Scholar - Asa01.M. Asada. The faithfulness of the monodromy representations associated with certain families of algebraic curves.
*J. Pure Appl. Algebra*, 159(2-3):123–147, 2001.CrossRefMATHMathSciNetGoogle Scholar - Bog09.M. Boggi. The congruence subgroup property for the hyperelliptic modular group: the open surface case.
*Hiroshima Math. J.*, 39(3):351–362, 2009.MATHMathSciNetGoogle Scholar - DM69.P. Deligne and D. Mumford. The irreducibility of the space of curves of given genus.
*Inst. Hautes Études Sci. Publ. Math.*, 36:75–109, 1969.CrossRefMATHMathSciNetGoogle Scholar - HM11.Y. Hoshi and S. Mochizuki. On the combinatorial anabelian geometry of nodally nondegenerate outer representations. to appear in Hiroshima Math. J.Google Scholar
- Hos09.Y. Hoshi. Absolute anabelian cuspidalizations of configuration spaces of proper hyperbolic curves over finite fields.
*Publ. Res. Inst. Math. Sci.*, 45(3):661–744, 2009.CrossRefMATHMathSciNetGoogle Scholar - Hos10.Y. Hoshi. Existence of nongeometric pro-
*p*Galois sections of hyperbolic curves.*Publ. Res. Inst. Math. Sci.*, 46(4):829–848, 2010.MATHMathSciNetGoogle Scholar - Hos11.Y. Hoshi. Galois-theoretic characterization of isomorphism classes of monodromically full hyperbolic curves of genus zero. to appear in Nagoya Math. J.Google Scholar
- Knu83.F. Knudsen. The projectivity of the moduli space of stable curves. II. The stacks
*M*_{g, n}.*Math. Scand.*, 52(2):161–199, 1983.Google Scholar - Mat11.M. Matsumoto. Difference between Galois representations in automorphism and outer-automorphism groups of a fundamental group.
*Proc. Amer. Math. Soc.*, 139(4):1215–1220, 2011.CrossRefMATHMathSciNetGoogle Scholar - Moc98.S. Mochizuki. Correspondences on hyperbolic curves.
*J. Pure Appl. Algebra*, 131(3):227–244, 1998.CrossRefMATHMathSciNetGoogle Scholar - Moc99.S. Mochizuki. The local pro-
*p*anabelian geometry of curves.*Invent. Math.*, 138(2):319–423, 1999.CrossRefMATHMathSciNetGoogle Scholar - Moc03.S. Mochizuki. The absolute anabelian geometry of canonical curves.
*Doc. Math.*, Extra Volume: Kazuya Kato’s fiftieth birthday:609–640, 2003.Google Scholar - Moc04.S. Mochizuki. The absolute anabelian geometry of hyperbolic curves.
*Galois theory and modular forms*, pages 77–122, 2004.Google Scholar - Moc10.S. Mochizuki. On the combinatorial cuspidalization of hyperbolic curves.
*Osaka J. Math*, 47(3):651–715, 2010.MATHMathSciNetGoogle Scholar - MT00.M. Matsumoto and A. Tamagawa. Mapping class group action versus Galois action on profinite fundamental groups.
*Amer. J. Math.*, 122(5):1017–1026, 2000.CrossRefMATHMathSciNetGoogle Scholar - MT08.S. Mochizuki and A. Tamagawa. The algebraic and anabelian geometry of configuration spaces.
*Hokkaido Math. J.*, 37(1):75–131, 2008.MATHMathSciNetGoogle Scholar - NT98.H. Nakamura and N. Takao. Galois rigidity of pro-
*ℓ*pure braid groups of algebraic curves.*Trans. Amer. Math. Soc*, 350(3):1079–1102, 1998.CrossRefMATHMathSciNetGoogle Scholar - SGA1.A. Grothendieck.
*Revêtements étale et groupe fondamental (SGA 1)*. Séminaire de géométrie algébrique du Bois Marie 1960-61, directed by A. Grothendieck, augmented by two papers by Mme M. Raynaud,*Lecture Notes in Math*. 224, Springer-Verlag, Berlin-New York, 1971. Updated and annotated new edition:*Documents Mathématiques*3, Société Mathématique de France, Paris, 2003.Google Scholar - SGA7-I.A. Grothendieck, M. Raynaud, and D. S. Rim.
*Groupes de monodromie en géométrie algébrique. I*. Lecture Notes in Mathematics, Vol. 288. Springer, 1972. Séminaire de Géométrie Algébrique du Bois-Marie 1967–1969 (SGA 7 I).Google Scholar - Tam97.A. Tamagawa. The Grothendieck conjecture for affine curves.
*Compositio Math*, 109(2):135–194, 1997.CrossRefMATHMathSciNetGoogle Scholar