Absolute Galois Group and Dessin d'Enfants
by
George Shabat
Russian State University of Humanities and Instute of Theoretical and Experimental Physics, Moscow, Russia.

Following Grothendieck. we call a ``dessins d'enfant'' any graph on the closed oriented surface, the complement to which is homeomorphic to a disjoint union of open cells. The construction of Grothendieck-Belyi defines the action of the absolute Galois group on the set of isomorphy classes of dessins d'enfants. This action is in a proper sense EXACT and therefore gives a unique opportunity of the VISUALIZATION of the Galois group. The talk will be devoted to the following aspects of this constructions.

The arithmetical Galois theory is intertwined with a so-called CARTOGRAPHICAL one. The examples will be given of the visible Galois invariants of the dessins obtained this way.

Even very particular cases of the dessins theory (namely, plane trees of diameter 4 and of small central valencies) lead to fascinating arithmetical problems related to the splitting of the Galois orbits. The brief overview (including some recent results of the speaker) will be presented.

Date received: January 31, 2001

Copyright © 2001 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cagi-16.