Action of the Grothendieck-Teichmüller group on torsion elements of full Teichmüller modular groups in genus zero

Collas, Benjamin

Journal de théorie des nombres de Bordeaux, Tome 24 (2012) no. 3, pp. 605-622.

Cet article présente l’action du groupe de Grothendieck-Teichmüller $\widehat{GT}$ sur les éléments de torsion d’ordre premier du groupe fondamental profini ${\pi }_1^{geom}\left({ℳ}_{0,\left[n\right]}\right)$. Nous établissons par ailleurs que les classes de conjugaison d’éléments de torsion d’ordre premier de ${\widehat{\pi }}_1\left({ℳ}_{0,\left[n\right]}\right)$ correspondent aux classes de conjugaison discrètes de ${\pi }_1\left({ℳ}_{0,\left[n\right]}\right)$.

In this paper we establish the action of the Grothendieck-Teichmüller group $\widehat{GT}$ on the prime order torsion elements of the profinite fundamental group ${\pi }_1^{geom}\left({ℳ}_{0,\left[n\right]}\right)$. As an intermediate result, we prove that the conjugacy classes of prime order torsion of ${\widehat{\pi }}_1\left({ℳ}_{0,\left[n\right]}\right)$ are exactly the discrete prime order ones of the ${\pi }_1\left({ℳ}_{0,\left[n\right]}\right)$.

[Bro74] Kenneth S. Brown, Euler characteristics of discrete groups and $G$-spaces. Invent. Math. 27 (1974), 229–264. | MR | Zbl

[Bro94] —, Cohomology of groups. Graduate Texts in Mathematics, Springer-Verlag, 1994. | MR

[Col11a] Benjamin Collas, Action of a Grothendieck-Teichmüller group on torsion elements of full Teichmüller modular groups of genus one. International Journal of Number Theory 8 (2012), no. 3, 763–787. | MR

[Col11b] —, Groupes de Grothendieck-Teichmüller et inertie champêtre des espaces de modules de genre zéro et un. Ph.D. thesis, Institut de Mathématiques de Jussieu, 2011.

[Dri90] V. G. Drinfel′d, On quasitriangular quasi-Hopf algebras and on a group that is closely connected with $Gal(\overline{\mathbb{Q}}/\mathbb{Q})$. Algebra i Analiz 2 (1990), no. 4, 149–181. | MR | Zbl

[Gro97] Alexandre Grothendieck, Esquisse d’un programme. Geometric Galois Actions I (Pierre Lochak and Leila Schneps, eds.), vol. 242, 1997, pp. 5–48. | MR | Zbl

[Hue79] J. Huebschmann, Cohomology theory of aspherical groups and of small cancellation groups. Journal of Pure Applied Algebra 14 (1979), 137–143. | MR | Zbl

[Iha94] Yasutaka Ihara, On the embedding of $\mathrm{Gal}(\overline{\mathbf{Q}}/\mathbf{Q})$ into $\widehat{\mathrm{GT}}$. The Grothendieck theory of dessins d’enfants (Luminy, 1993), London Math. Soc. Lecture Note Ser., vol. 200, Cambridge Univ. Press, Cambridge, 1994, With an appendix: the action of the absolute Galois group on the moduli space of spheres with four marked points by Michel Emsalem and Pierre Lochak, pp. 289–321. | MR | Zbl

[IM95] Yasutaka Ihara and Makoto Matsumoto, On Galois actions of profinite completions of braid groups. Recent Developments in the inverse Galois problem, M. Fried et al. Eds., AMS, 1995. | MR | Zbl

[Ker83] Steven P. Kerckhoff, The Nielsen realization problem. Ann. of Math. (2) 117 (1983), no. 2, 235–265. | MR | Zbl

[LS94] Pierre Lochak and Leila Schneps, The Grothendieck-Teichmüller group as automorphisms of braid groups. The Grothendieck theory of dessins d’Enfants, L. Schneps, London Mathematical Society, 1994. | Zbl

[LS97a] —, A cohomological interpretation of the Grothendieck-Teichmüller group. Inventiones mathematica (1997), no. 127, 571–600. | MR

[LS97b] —, The universal Ptolemy-Teichmüller groupoid. Geometric Galois actions, 2, London Math. Soc. Lecture Note Ser., vol. 243, Cambridge Univ. Press, Cambridge, 1997, pp. 325–347.

[LSN04] Pierre Lochak, Leila Schneps, and Hiroaki Nakamura, Eigenloci of 5 point configurations of the Riemann sphere and the Grothendieck-Teichmüller group. Mathematical Journal of Okayama University 46 (2004), 39–75. | MR | Zbl

[MH75] C. Maclachlan and W. J. Harvey, On mapping class groups and Teichmüller spaces. Proc. London Math. Soc. 30 (1975), no. 3, 496–512. | MR | Zbl

[Nak94] Hiroaki Nakamura, Galois rigidity of pure sphere braid groups and profinite calculus. Journal Mathematical Sciences University Tokyo 1 (1994), 71–136. | MR | Zbl

[Nak97] —, Galois representations in the profinite Teichmüller modular groups. London Math. Soc. Lecture Note Series 242 (1997), 159–174.

[Nak99] —, Limits of Galois representations in fundamental groups along maximal degeneration of marked curves. I. Amer. J. Math. 121 (1999), no. 2, 315–358. | MR

[NT03] Hiroaki Nakamura and Hiroshi Tsunogai, Harmonic and equianharmonic equations in the Grothendieck-Teichmüller group.. Forum Math. 15 (2003), no. 6, 877–892. | MR | Zbl

[NTY10] Hiroaki Nakamura, Hiroshi Tsunogai and Seidai Yasuda, Harmonic and equianharmonic equations in the Grothendieck–Teichmüller group. III. Journal of the Institute of Mathematics of Jussieu 9 (2010), no. 2, 431–448. | MR | Zbl

[Noo04] B. Noohi, Fundamental groups of algebraic stacks. Journal of the Institute of Mathematics of Jussieu 3 (2004), no. 01, 69–103. | MR | Zbl

[NS03] Nikolay Nikolov and Dan Segal, Finite index subgroups in profinite groups.. Comptes Rendus Mathematique 337 (2003), no. 5, 303–308. | MR | Zbl

[Oda97] Takayuki Oda, Etale homotopy type of the moduli spaces of algebraic curves. Geometric Galois actions, 1, London Math. Soc. Lecture Note Ser., vol. 242, Cambridge Univ. Press, Cambridge, 1997, pp. 85–95. | MR | Zbl

[Sch97] Claus Scheiderer, Appendix to: A cohomological interpretation of the Grothendieck-Teichmüller group. Inventiones mathematica 127 (1997), 597–600. | MR

[Sch03] Leila Schneps, Special loci in moduli spaces of curves. Galois groups and fundamental groups, vol. 41, Cambridge University Press, 2003. | MR | Zbl

[Sch06] —, Automorphisms of curves and their role in Grothendieck-Teichmüller theory. Mathematische Nachrichten 279 (2006), no. 5-6, 656–671. | MR

[Ser94] Jean-Pierre Serre, Cohomologie galoisienne. Lecture Notes in Mathematics, Springer-Verlag, 1994. | MR | Zbl

[Ser97] —, Two letters on non-abelian cohomology. Geometric galois actions: around Grothendieck’s esquisse d’un programme, Cambridge University Press, 1997.

[Ser03] —, Trees. Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2003, Translated from the French original by John Stillwell, Corrected 2nd printing of the 1980 English translation. | MR

[Tsu06] Hiroshi Tsunogai, Some new-type equations in the Grothendieck-Teichmüller group arising from geometry of ${ℳ}_{0,5}$. Primes and knots, Contemp. Math., vol. 416, Amer. Math. Soc., Providence, RI, 2006, pp. 263–284. | MR | Zbl