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Galois rigidity of pro-$l$ pure braid groups
of algebraic curves

Authors: Hiroaki Nakamura and Naotake Takao
Journal: Trans. Amer. Math. Soc. 350 (1998), 1079-1102
MSC (1991): Primary 14E20; Secondary 20F34, 20F36
MathSciNet review: 1443885
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Abstract: In this paper, Grothendieck's anabelian conjecture on the pro-$l$ fundamental groups of configuration spaces of hyperbolic curves is reduced to the conjecture on those of single hyperbolic curves. This is done by estimating effectively the Galois equivariant automorphism group of the pro-$l$ braid group on the curve. The process of the proof involves the complete determination of the groups of graded automorphisms of the graded Lie algebras associated to the weight filtration of the braid groups on Riemann surfaces.

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  • [A] M.Asada, On the filtration of topological and pro-$l$ mapping class groups of punctured Riemann surfaces, J. Math. Soc. Japan 48 (1996), 13-36. CMP 96:04
  • [AN] M.Asada, H.Nakamura, On graded quotient modules of mapping class groups of surfaces, Israel J. Math. 90 (1995), 93-113. MR 96j:57015
  • [Bog] F.A.Bogomolov, Points of finite order on an abelian variety, Math. USSR Izvestiya 17 (1981), 55-72. MR 81m:14031
  • [Bor] A.Borel, Linear Algebraic Groups, Second enlarged edition, (1st edition from Benjamin 1969), Springer, 1991. MR 92d:20001
  • [BK] A.K.Bousfield, D.M.Kan, Homotopy limits, completions and localizations, Lect. Notes in Math., vol. 304, Springer, 1972. MR 51:1825
  • [De] P.Deligne, Le groupe fondamental de la droite projective moins trois points, The Galois Group over $Q$, ed. by Y.Ihara, K.Ribet, J.-P.Serre, Springer, 1989, pp. 79-297. MR 90m:14016
  • [Di] J.Dieudonné, Sur les groupes classiques, Actual. Scient. et Ind., n${}^{\circ }$ 1040, Hermann, Paris, 1948. MR 9:494c
  • [DSMS] J.D.Dixon, M.P.F.du Sautoy, A.Mann, D.Segal, Analytic pro-$p$ groups, London Math. Soc. Lect. Notes Series, vol. 157, Cambridge Univ. Press, 1991. MR 94e:20037
  • [G1] A.Grothendieck, Letter to G.Faltings, 1983.
  • [G2] A.Grothendieck, Esquisse d'un Programme, mimeographed note (1984).
  • [G3] A.Grothendieck, Revêtement Etales et Groupe Fondamental (SGA1), Lecture Notes in Math. 224 (1971). MR 50:7129
  • [Hir] K.A.Hirsch, On infinite soluble groups II, Proc. London Math. Soc. (2) 44 (1933), 336-344.
  • [Ih] Y.Ihara, Automorphisms of pure sphere braid groups and Galois representations, The Grothendieck Festschrift, Volume II, Birkhäuser, 1990, pp. 353-373. MR 92k:20077
  • [IhK] Y.Ihara, M.Kaneko, Pro-$l$ pure braid groups of Riemann surfaces and Galois representations, Osaka J. Math. 29 (1992), 1-19. MR 93i:14022
  • [Iv] N.V.Ivanov, Algebraic properties of mapping class groups of surfaces, Geometric and Algebraic Geometry., Banach Center Publ., 1986, pp. 15-35. MR 89a:57009
  • [K] M.Kaneko, Certain automorphism groups of pro-$l$ fundamental groups of punctured Riemann surfaces., J. Fac. Sci. Univ. Tokyo 36 (1989), 363-372. MR 90j:14019
  • [KT] Y.Kawahara, T.Terasoma, Galois-Torelli theorems for hyperplane arrangements, in preparation.
  • [Ma] M.Matsumoto, On Galois representations on profinite braid groups of curves, J. reine angew. Math. 474 (1996), 169-219. CMP 96:13
  • [M1] S.Mochizuki, The profinite Grothendieck conjecture for closed hyperbolic curves over number fields, J. Math. Sci., Univ. Tokyo 3 (1996), 571-627. CMP 97:07
  • [M2] S.Mochizuki, The local pro-$p$ Grothendieck conjecture for hyperbolic curves, Preprint RIMS-1045 (1995), RIMS-1097 (1996).
  • [N1] H.Nakamura, On Galois automorphisms of the fundamental group of the projective line minus three points, Math. Z. 206 (1991), 617-622. MR 92i:14014
  • [N2] -, Galois rigidity of pure sphere braid groups and profinite calculus, J. Math. Sci., Univ. Tokyo 1 (1994), 71-136. MR 96e:14021
  • [N3] -, Galois rigidity of algebraic mappings into some hyperbolic varieties, Internat. J. Math. 4 (1993), 421-438. MR 94f:14011
  • [N4] -, On exterior Galois representations associated with open elliptic curves, J. Math. Sci., Univ. Tokyo 2 (1995), 197-231. MR 97a:11086
  • [N5] -, Coupling of universal monodromy representations of Galois- Teichmüller modular groups, Math. Ann. 304 (1996), 99-119. MR 97a:14026
  • [NT] -, H.Tsunogai, Some finiteness theorems on Galois centralizers in pro-$l$ mapping class groups, J. reine angew Math. 441 (1993), 115-144. MR 94g:14005
  • [NTU] -, N.Takao and R.Ueno, Some stability properties of Teichmüller modular function fields with pro-$l$ weight structures, Math. Ann. 302 (1995), 197-213. MR 96h:14041
  • [O] T.Oda, Galois actions on the nilpotent completion of the fundamental group of an algebraic curve, Advances in Number Theory, F.Q.Gouvêa, N.Yui eds., Clarendon Press, Oxford, 1993, pp. 213-232. CMP 96:06
  • [P] F.Pop, On Grothendieck's conjecture of birational anabelian geometry, Ann. of Math. 138 (1994), 145-182; Part 2, Ann. of Math. (to appear). MR 94m:14007
  • [Q] D.Quillen, Rational homotopy Theory, Ann. of Math. 90 (1969), 205-295. MR 41:2678
  • [Sc] G.P.Scott, Braid groups and the group of homeomorphisms of a surface, Proc. Camb. Phil. Soc. 68 (1970), 605-617. MR 42:3786
  • [Se] J.P.Serre, Cohomologie Galoisienne, 4th ed., Lecture Notes in Math. 5 (1973). MR 53:8030
  • [Tk] N.Takao, Braid monodromies on proper curves and Galois representations, in preparation.
  • [Tm] A.Tamagawa, The Grothendieck conjecture for affine curves, Compositio Math. (to appear).
  • [W] R.B.Warfield, Jr, Nilpotent Groups, Lect. Notes in Math., vol. 513, Springer, 1976. MR 53:13413

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Additional Information

Hiroaki Nakamura
Affiliation: Department of Mathematical Sciences, University of Tokyo, Komaba, Meguro-ku, Tokyo 153, Japan

Naotake Takao
Affiliation: Research Institute for Mathematical Sciences, Kyoto University, Kitashirakawa, Kyoto 606-01, Japan

Keywords: Galois representation, anabelian geometry, braid group
Received by editor(s): September 10, 1995
Article copyright: © Copyright 1998 American Mathematical Society

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