Dessin d'enfant

introduction

• Grothendieck's theory of dessins provides a bridge between algebraic numbers and combinatorics
• a dessin is essentially a bipartite graph embedded on a compact, oriented surface (without boundary), and that the absolute Galois group \(\operatorname{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})\) acts on (isomorphism classes of) dessins.
• Grothendieck-Teichmüller group of a finite group \(G\), denoted \(\mathcal{GT}(G)\)
• there is an action of \(\mathcal{GT}(G)\) on those dessins whose monodromy group is \(G\), and the Galois action on the same objects factors via a map \(\operatorname{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}) \longrightarrow \mathcal{GT}(G)\)
• Motivation for the study of all groups \(\mathcal{GT}(G)\), for all groups \(G\), is increased by the fact that the combined map

\[ \operatorname{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}) \longrightarrow \mathcal{GT} := \lim_G \, \mathcal{GT}(G) \] is injective.

memo

• http://www.neverendingbooks.org/index.php/kleins-dessins-denfant-and-the-buckyball.html
• http://www.neverendingbooks.org/index.php/permutation-representations-of-monodromy-groups.html

related items

• Grothendieck-Teichmuller theory

computational resource

• https://github.com/vbeffara/Simulations

encyclopedia

• http://en.wikipedia.org/wiki/Dessin_d'enfant

books

• Guralnick, Robert M., and John Shareshian. 2007. Symmetric and Alternating Groups as Monodromy Groups of Riemann Surfaces I: Generic Covers and Covers with Many Branch Points. American Mathematical Soc.
• Schneps, Leila, ed. 1994. The Grothendieck Theory of Dessins D’enfants. Vol. 200. London Mathematical Society Lecture Note Series. Cambridge: Cambridge University Press. http://www.ams.org/mathscinet-getitem?mr=1305390.

expositions

• Norbert A'Campo, Lizhen Ji, Athanase Papadopoulos, On Grothendieck's tame topology, http://arxiv.org/abs/1603.03016v1
• Cueto, Moises Herradon. “The Field of Moduli and Fields of Definition of Dessins D’enfants.” arXiv:1409.7736 [math], September 26, 2014. http://arxiv.org/abs/1409.7736.
• Planat, Michel. “Drawing Quantum Contextuality with ‘Dessins D’enfants’.” arXiv:1404.6986 [math-Ph, Physics:quant-Ph], April 28, 2014. http://arxiv.org/abs/1404.6986.
• Jones, Dessins d'enfants: bipartite maps and Galois groups
• Eriksson, Galois theory of Covers
• Robalo, Galois Theory towards Dessins d'Enfants, masters thesis
• Introduction to Dessins d'Enfants, undergraduate research program

articles

• Pierre Guillot, The Grothendieck-Teichmüller group of \(PSL(2, q)\), arXiv:1604.04415 [math.GR], April 15 2016, http://arxiv.org/abs/1604.04415
• Khashayar Filom, Ali Kamalinejad, Dessins on Modular Curves, http://arxiv.org/abs/1603.01693v1
• Planat, Michel, and Hishamuddin Zainuddin. “Zoology of Atlas-Groups: Dessins D’enfants, Finite Geometries and Quantum Commutation.” arXiv:1601.04865 [math-Ph, Physics:quant-Ph], January 19, 2016. http://arxiv.org/abs/1601.04865.
• Hilany, Boulos El. “Counting Positive Intersection Points of a Trinomial and a \(\mathbf{T}\)-Nomial Curves via Groethendieck’s Dessin D’enfant.” arXiv:1512.05688 [math], December 17, 2015. http://arxiv.org/abs/1512.05688.
• Wang, Na-Er, Roman Nedela, and Kan Hu. “Totally Symmetric Dessins with Nilpotent Automorphism Groups of Class Three.” arXiv:1511.06863 [math], November 21, 2015. http://arxiv.org/abs/1511.06863.
• Cueto, Moisés Herradón. “An Explicit Quasiplatonic Curve with Field of Moduli \(\mathbb Q(\sqrt[3]{2})\).” arXiv:1509.05819 [math], September 18, 2015. http://arxiv.org/abs/1509.05819.
• Fine, Jonathan 2015Bias and Dessins. arXiv:1506.06389 [math]. http://arxiv.org/abs/1506.06389, accessed July 11, 2015.
• Beffara, Vincent. ‘Dessins D’enfants for Analysts’. arXiv:1504.00244 [math], 1 April 2015. http://arxiv.org/abs/1504.00244.
• Bose, Sownak, James Gundry, and Yang-Hui He. “Gauge Theories and Dessins d’Enfants: Beyond the Torus.” arXiv:1410.2227 [hep-Th], October 8, 2014. http://arxiv.org/abs/1410.2227.
• Guillot, Pierre. 2014. “Some Computations with the Grothendieck-Teichm"uller Group and Equivariant Dessins D’enfants.” arXiv:1407.3112 [math], July. http://arxiv.org/abs/1407.3112.
• Kazarian, Maxim, and Peter Zograf. 2014. “Virasoro Constraints and Topological Recursion for Grothendieck’s Dessin Counting.” arXiv:1406.5976 [math], June. http://arxiv.org/abs/1406.5976.

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