Fuchsian triangle groups and Grothendieck dessins. Variations on a theme of Belyi
type: Item
instance of: scholarly article
title: Fuchsian triangle groups and Grothendieck dessins. Variations on a theme of Belyi
zbMATH Open document ID: 0811.14030
DOI: 10.1007/BF02101464
author: Paula Tretkoff
author: C. Itzykson
author: Jürgen Wolfart
published in: Communications in Mathematical Physics
publication date: 1994-08-10T00:00:00Z
review text: According to a theorem of Belyi, a smooth projective algebraic curve \(X\) is defined over a number field if and only if there is a non-constant element of its function field ramified only over 0, 1, and \(\infty\). The existence of such a Belyi function \(\beta\) is equivalent to that of a representation of the curve as a possibly compactified quotient of the Poincaré upper half plane by a subgroup of finite index in a Fuchsian triangle group. If in particular \(\beta\) ramifies in all points of \(\beta^{-1}(1)\) with ramification order 2, the graph \(\beta^{-1} ([0,1])\) on \(X\) is called a ``Grothendieck dessin'', and it is shown that \(X\) is uniquely determined up to isomorphisms already by the topology of its Grothendieck dessins. Examples such as the Platonic solids on the sphere and natural triangulations of the Fermat curves are discussed in detail, the interplay with combinatorial objects as the ``cartographic group'' of the dessin is described, and connections with Shimura varieties and parts of physics like rational triangular billards are indicated.
Mathematics Subject Classification ID: 14H30
Mathematics Subject Classification ID: 30F10
Mathematics Subject Classification ID: 30F35
Mathematics Subject Classification ID: 20B25
zbMATH DE Number: 617384
zbMATH Keywords: Grothendieck dessin
zbMATH Keywords: cartographic group
zbMATH Keywords: Belyi function
zbMATH Keywords: Fuchsian triangle group
zbMATH Keywords: platonic solids
zbMATH Keywords: Shimura varieties
zbMATH Keywords: rational triangular billards
reviewed by: Jürgen Wolfart
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 617384