Hecke Groups, Dessins d'Enfants and the Archimedean Solids
Abstract
Grothendieck's dessins d'enfants arise with everincreasing frequency in many areas of 21st century mathematical physics. In this paper, we review the connections between dessins and the theory of Hecke groups. Focussing on the restricted class of highly symmetric dessins corresponding to the socalled Archimedean solids, we apply this theory in order to provide a means of computing representatives of the associated conjugacy classes of Hecke subgroups in each case. The aim of this paper is to demonstrate that dessins arising in mathematical physics can point to new and hitherto unexpected directions for further research. In addition, given the particular ubiquity of many of the dessins corresponding to the Archimedean solids, the hope is that the computational results of this paper will prove useful in the further study of these objects in mathematical physics contexts.
 Publication:

Frontiers in Physics
 Pub Date:
 December 2015
 DOI:
 10.3389/fphy.2015.00091
 arXiv:
 arXiv:1309.2326
 Bibcode:
 2015FrP.....3...91H
 Keywords:

 Dessins d'enfants;
 Archimedean solids;
 Platonic solids;
 Belyi maps;
 Hecke groups;
 Mathematics  Algebraic Geometry;
 High Energy Physics  Theory;
 Mathematics  Number Theory
 EPrint:
 28 pages, 1 figure. v2: Substantial streamlining. v3: Minor changes