In mathematics, the Grothendieck–Teichmüller group GT is a group closely related to (and possibly equal to) the absolute Galois group of the rational numbers. It was introduced by Vladimir Drinfeld (1990) and named after Alexander Grothendieck and Oswald Teichmüller, based on Grothendieck's suggestion in his 1984 essay Esquisse d'un Programme to study the absolute Galois group of the rationals by relating it to its action on the Teichmüller tower of Teichmüller groupoids Tg,n, the fundamental groupoids of moduli stacks of genus g curves with n points removed.
There are several variations of the group:
- a pro-l version, which is motivic [1]
- a k-pro-unipotent version, and
- a profinite version, which is anabelian [2],[3];
These versions were jointly defined by V. Drinfeld and Y.Ihara.
References
editGeneral references
edit- Collas, Benjamin (2026-03-03). "Anabelian perspectives in Galois-Teichmüller theory". arXiv:2603.02848 [math.AG].
- Drinfeld, V. G. (1990), "On quasitriangular quasi-Hopf algebras and on a group that is closely connected with Gal(Q/Q)", Rossiĭskaya Akademiya Nauk. Algebra i Analiz (in Russian), 2 (4): 149–181, ISSN 0234-0852, MR 1080203 Translation in Leningrad Math. J. 2 (1991), no. 4, 829–860.
- Ihara, Yasutaka (1990). "Braids, Galois groups, and some arithmetic functions". Proceedings of the International Congress of Mathematicians. Vol. 1–2. Kyoto, Japan: Mathematical Society of Japan (published 1991). pp. 99–120.
- Schneps, Leila (1997), "The Grothendieck–Teichmüller group GT: a survey", in Schneps, Leila; Lochak, Pierre (eds.), Geometric Galois actions, 1 (PDF), London Math. Soc. Lecture Note Ser., vol. 242, Cambridge University Press, pp. 183–203, doi:10.1017/CBO9780511666124, ISBN 978-0-521-59642-8, MR 1483118
Further reading
edit- Fresse, Benoit (2017), Homotopy of Operads and Grothendieck-Teichmüller Groups: Part 2: The Applications of (Rational) Homotopy Theory Methods, Mathematical Surveys and Monographs, vol. 217, American Mathematical Society, p. 704, ISBN 9781470434823
Relation to combinatorial anabelian geometry
edit- Hoshi, Yuichiro; Minamide, Arata; Mochizuki, Shinichi (2022). "Group-theoreticity of numerical invariants and distinguished subgroups of configuration space groups". Kodai Mathematical Journal. 45 (3): 295-348. doi:10.2996/kmj45301.
- Hoshi, Yuichiro; Mochizuki, Shinichi; Tsujimura, Shota (2025). "Combinatorial construction of the absolute Galois group of the field of rational numbers". Journal of Mathematical Sciences, the University of Tokyo. 32 (1): 1–125.