Grothendieck–Teichmüller group

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 T_g,n, the fundamental groupoids of moduli stacks of genus g curves with n points removed. There are several minor variations of the group: a discrete version, a pro-l version, a k-pro-unipotent version, and a profinite version; the first three versions were defined by Drinfeld, and the version most often used is the profinite version.

References

General references

• 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.
• 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

Relation to algebraic topology via the little disks operads

• 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

• 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.