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

Abstract

In this article we study the action of the absolute Galois group Gal(Q/Q) on dessins d’enfants and Beauville surfaces. A foundational result in Grothendieck’s theory of dessins d’enfants is the fact that the absolute Galois group Gal(Q/Q) acts faithfully on the set of all dessins. However the question of whether this holds true when the action is restricted to the set of the, more accessible, regular dessins seems to be still an open question. In the first part of this paper we give an affirmative answer to it. In fact we prove the strongest result that the action is faithful on the set of quasiplatonic (or triangle) curves of any given hyperbolic type. Beauville surfaces are an important kind of algebraic surfaces introduced by Catanese. They are rigid surfaces of general type closely related to dessins d’enfants. Here we prove that for any σ ∈ Gal(Q/Q) different from the identity and the complex conjugation there is a Beauville surface S such that S and its Galois conjugate S have non-isomorphic fundamental groups. This in turn easily implies that the action of Gal(Q/Q) on the set of Beauville surfaces is faithful. These results were conjectured by Bauer, Catanese and Grunewald, and immediately imply that Gal(Q/Q) acts faithfully on the connected components of the moduli space of surfaces of general type, a result due to the above mentioned authors.