A note on the action of the absolute Galois group on dessins

Abstract

We show that the action of the absolute Galois group on dessins d'enfants of given genus g is faithful, a result that had been previously established for g = 0 and g = 1.

Bipartite graph embeddings, Riemann surfaces and Galois groups

“…There is a natural action of G on curves defined over Q, and hence on the associated dessins, such as bipartite maps. As observed by Grothendieck, this action is faithful, so that in a sense one can 'see' the whole of G through its action on very simple combinatorial objects; this remains true even when the action is restricted to apparently small and accessible classes of dessins such as plane trees or maps of a given genus, as proved by Schneps [59] and by Girondo and González-Diez [26].…”

“…There is a natural action of G on Belyȋ pairs, obtained from its action on the coefficients of the polynomials and rational functions defining them, so it has an induced action on the associated dessins. As shown by Streit and the author in [46], such properties as the numbers and valencies of vertices and faces, the genus, the monodromy group and the orientation-and colourpreserving automorphism group are invariant under G. Nevertheless, this action of G is faithful, even when restricted to such simple dessins as plane trees [59] or to dessins of a given genus [26]. In a sense, this allows one to 'see' the whole of the Galois theory of algebraic number fields through the action of G on dessins.…”

The absolute Galois group acts faithfully on regular dessins and on Beauville surfaces

“…Grothendieck noted that the action of Gal(Q/Q) is faithful already on dessins of genus 1. Later, it was shown that it is also faithful on dessins of any given genus [14,33] (see also [15]). In this article, we prove the following two theorems relative to this action, the second one being a stronger form of Conjecture 2.13 in Catanese's survey article [8] ( [4,Conjecture 4.10]).…”

Regular dessins d'enfants with dicyclic group of automorphisms

“…This process provides of a natural equivalence between (isomorphism classes of) Belyi pairs and (isomorphism classes of) dessins d'enfants and, by Belyi's theorem, there is a natural action of the absolute Galois group Gal(Q/Q) on (isomorphism classes of) dessins d'enfants. It is well known that this action is faithfull [11,13,21]. Associated to a Belyi pair (S , β) (respectively, a dessin d'enfant D = (X, G)) is its group Aut + (S , β) (respectively, Aut + (D)) consisting of those conformal automorphisms ϕ of S such that β = β • ϕ (respectively, the group of homotopy class of those orientation-preserving self-homeomorphisms of X keeping invariant G β and the colour of the vertices).…”