Finite tripod variants of I/OM

On Ihara’s question/Oda–Matsumoto conjecture
  • Florian PopEmail author


In this note we introduce and prove a wide generalization and sharpening of Ihara’s question / Oda–Matsumoto conjecture, for short I/OM. That leads to a quite concrete topological/combinatorial description of absolute Galois groups, in particular of \(\mathrm{Gal}_{{\mathbb {Q}}}=\mathrm{Aut}(\overline{{\mathbb {Q}}})\), as envisioned by Grothendieck in his Esquisse d’ un Programme.

Mathematics Subject Classification

Primary 11G99 12F10 12G99 14A99 



I would like to thank Ching-Li Chai, Franz Oort, Jakob Stix, Alexander Schmidt and Tamás Szamuely for technical discussions and help, Pierre Lochak for insisting that these facts should be thoroughly investigated, and many others who showed interest in this work: Yves André, Pierre Deligne, R. Hain, Y. Ihara, Minhyong Kim, M. Matsumoto, N. Nakamura, M. Saidi, A. Tamagawa for discussions on several occasions. Special thanks are due to the University of Heidelberg, University of Bonn, and there MPI Bonn, for the excellent working conditions during my visits there as visiting scientist. Last but not least, many thanks to the referee, for the careful reading of the manuscript and suggestions to improve the presentation.


  1. 1.
    André, Y.: On a geometric description of \({\rm Gal}(\overline{\mathbb{Q}}_p|{\mathbb{Q}})\) and a \(p\)-adic avatar of \(\widehat{GT}\). Duke Math. J. 119, 1–39 (2003)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Belyi, G.V.: On Galois extensions of a maximal cyclotomic field. Math. USSR Izv. 14(2), 247–256 (1980). (Original: Izvestiya Akademii Nauk SSSR, vol. 14(2), 269–276 (1979))Google Scholar
  3. 3.
    Bogomolov, F.A.: On two conjectures in birational algebraic geometry. In: Fujiki, A., et al. (eds.) Algebraic Geometry and Analytic Geometry, ICM-90 Satellite Conference Proceedings. Springer, Tokyo (1991)Google Scholar
  4. 4.
    Deligne, P.: Le groupe fondamental de la droite projective moins trois points. In: Galois Groups Over \({\bf Q},\) Mathematical Sciences Research Institute Publications, vol. 16, pp. 79–297, Springer, Berlin (1989)Google Scholar
  5. 5.
    Drinfeld, V.G.: On quasi-triangular quasi-Hopf algebras and on a group that is closely connected with \({\rm Gal}(\overline{\mathbb{Q}}/{\mathbb{Q}})\). Leningr. Math. J. 2(4), 829–860 (1991). Original: Algebra i Analiz 2(4), 149–181 (1990)Google Scholar
  6. 6.
    Furusho, H.: Geometric and arithmetic subgroups of the Grothendieck–Teichmüller group. Math. Res. Lett. 10, 97–108 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Furusho, H.: Multiple zeta values and Grothendieck–Teichmüller groups. AMS Contemp. Math. 416, 49–82 (2006)CrossRefzbMATHGoogle Scholar
  8. 8.
    Geometric Galois Actions I, LMS LNS vol. 242, Schneps, L., Lochak, P. (eds.). Cambridge University Press, Cambridge (1998)Google Scholar
  9. 9.
    Grothendieck, A.: Letter to Faltings, (June 1983), See [8]Google Scholar
  10. 10.
    Grothendieck, A.: Esquisse d’un programme (1984). See [8]Google Scholar
  11. 11.
    Hain, R., Matsumoto, M.: Tannakian fundamental groups associated to Galois groups. In: Schneps, L. (ed.) Galois Groups and Fundamental Groups, MSRI Publication Series, vol. 41, pp. 183–216 (2003)Google Scholar
  12. 12.
    Harbater, D., Schneps, L.: Fundamental groups of moduli and the Grothendieck–Teichmüller group. Trans. Am. Math. Soc. 352, 3117–3148 (2000)CrossRefzbMATHGoogle Scholar
  13. 13.
    Hatcher, A., Lochak, P., Schneps, L.: On the Teichmüller tower of mapping class groups. J. Reine Angew. Math. 521, 1–24 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Hoshi, Y., Mochizuki, S.: On the combinatorial anabelian geometry of nodally nondegenerate outer representations. Hiroshima Math. J. 41(3), 275–342 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Ihara, Y.: On Galois represent. arising from towers of covers of \({\mathbb{P}}^1{\backslash } \{0,1,\infty \}\). Invent. Math. 86, 427–459 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Ihara, Y.: Braids, Galois groups, and some arithmetic functions. In: Proceedings of the ICM’90, vol. I, II, Mathematical Society Japan, Tokyo, pp. 99–120 (1991)Google Scholar
  17. 17.
    Ihara, Y., On the Embedding of \(\text{Gal}({\mathbb{Q}})\) into \(\widehat{GT}\), the Grothendieck Theory of Dessins d’enfants, (Luminy, 1993). LMS LNS, 200, pp. 289–321. Cambridge University Press, Cambridge (1994)Google Scholar
  18. 18.
    Ihara, Y., Matsumoto, M.: On Galois actions on profinite completions of braid groups. In: Recent Developments in the Inverse Galois Problem (Seattle, WA, 1993). Contemporary Mathematics, vol. 186, pp. 173–200. AMS, Providence (1995)Google Scholar
  19. 19.
    Lochak, P., Nakamura, H., Schneps, L.: On a new version of the Grothendieck–Teichmüller group. C. R. Acad. Sci. 325, 11–16 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Lochak, P., Nakamura, H., Schneps, L.: Eigenloci of 5 point configurations on the Riemann sphere and the Grothendieck–Teichmueller group. Math. J. Okayama Univ. 46, 39–75 (2004)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Lochak, P., Schneps, L.: A cohomological interpretation of the Grothendieck–Teichmüller group. Appendix by C. Scheiderer. Invent. Math. 127, 571–600 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Matsumoto, M.: Galois representations on profinite braid groups on curves. J. Reine Angew. Math. 474, 169–219 (1996)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Matsumoto, M., Tamagawa, A.: Mapping class-group action versus Galois action on profinite fundamental groups. Am. J. Math. 122, 1017–1026 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Nakamura, H.: Limits of Galois representations in fundamental groups along maximal degeneration of marked curves II. Proc. Symp. Pure Math. 70, 43–78 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Nakamura, H., Schneps, L.: On a subgroup of the Grothendieck–Teichmüller group acting on the profinite Teichmüller modular group. Invent. Math. 141, 503–560 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Pop, F.: Pro-l abelian-by-central Galois theory of Zariski prime divisors. Isr. J. Math. 180, 43–68 (2010)CrossRefzbMATHGoogle Scholar
  27. 27.
    Pop, F.: Inertia elements versus Frobenius elements. Math. Ann. 438, 1005–1017 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Pop, F.: Recovering fields from their decomposition graphs. In: Goldfeld, D., Jorgenson, J., Ramakrishnan, D., Ribet, K., Tate, J. (eds.) Number Theory, Analysis and Geometry—In Memory of Serge Lang, Springer Special Volume 2010, pp. 519–594Google Scholar
  29. 29.
    Pop, F.: On the birational anabelian program initiated by Bogomolov I. Invent. Math. 187, 511–533 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
  31. 31.
    Schneps, L.: Dessins d’enfants on the Riemann sphere. In: The Grothendieck Theory of Dessins Denfants (Luminy, 1993), LMS LNS, vol. 200, pp. 47–77. Cambridge University Press, Cambridge (1994)Google Scholar
  32. 32.
    Topaz, A.: Commuting-liftable subgroups of Galois groups II. J. Reine Angew. Math. 730, 65–133 (2017)MathSciNetzbMATHGoogle Scholar
  33. 33.
    Topaz, A.: On the Nature of Base Fields, An Appendix to: On the Minimized Decomposition Theory of Valuations, by F. Pop, Bulletin Mathematical Society Science Mathematics, Roumanie, Tome, vol. 58(106), No. 3, pp. 331–357 (2015)Google Scholar
  34. 34.
    Topaz, A.: The Galois action on geometric lattices and the Mod-\(\ell \) I/OM. Invent. Math. 213(2), 371–459 (2018)MathSciNetCrossRefzbMATHGoogle Scholar

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Authors and Affiliations

  1. 1.Department of MathematicsUniversity of PennsylvaniaPhiladelphiaUSA

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