mathematics

Grothendieck–Teichmüller group

Grothendieck–Teichmüller group is a mathematics topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Grothendieck–Teichmüller group rather than just read about it. In short: 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…

Key takeaways

Grothendieck–Teichmüller group belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Grothendieck–Teichmüller group to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Grothendieck–Teichmüller group from memory before moving on to harder problems.

Reference excerpt

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 a k-pro-unipotent version, and a profinite version, which is anabelian ,; These versions were jointly defined by V. Drinfeld and Y.Ihara.

References

General references 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

Worked examples

Example 1 — a first encounter with Grothendieck–Teichmüller group

Start with the simplest possible case. Write down what Grothendieck–Teichmüller group claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Grothendieck–Teichmüller group before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Grothendieck–Teichmüller group ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Grothendieck–Teichmüller group

In research: Grothendieck–Teichmüller group appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Grothendieck–Teichmüller group in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Grothendieck–Teichmüller group is common in secondary-school and first-year university syllabi. It links to neighbouring topics Galois theory, Number theory, so understanding it makes those chapters shorter.
In everyday life: Look for Grothendieck–Teichmüller group outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

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How to study Grothendieck–Teichmüller group in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Grothendieck–Teichmüller group means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Grothendieck–Teichmüller group out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Grothendieck–Teichmüller group in simple terms?

Why does Grothendieck–Teichmüller group matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Grothendieck–Teichmüller group?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Grothendieck–Teichmüller group.