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

Abstract

Beauville surfaces are an important kind of algebraic surfaces introduced by Catanese. They are rigid surfaces of general type defined over number fields. We prove that, for any σ∈Gal(double-struckQ¯/Q) different from the identity and the complex conjugation, there is a Beauville surface S such that S and its Galois conjugate Sσ have non‐isomorphic fundamental groups. This in turn easily implies that the action of Gal(double-struckQ¯/Q) on the set of isomorphism classes of Beauville surfaces is faithful. These results were conjectured by Bauer, Catanese and Grunewald, and immediately imply that Gal(double-struckQ¯/Q) acts faithfully on the connected components of the moduli space of surfaces of general type, a result due to the above‐mentioned authors. These results on Beauville surfaces heavily depend on the fact that the absolute Galois group acts faithfully on regular dessins, a result that we prove in this paper. Moreover, we show the stronger result that the action is faithful on the set of quasiplatonic (or triangle) curves of any given hyperbolic type.

Topological methods in moduli theory

“…Further developments have been announced in [183] by Gonzaléz-Diez and Jaikin-Zapirain: for instance the faithfulness of the action of the absolute Galois group on the discrete set of the moduli space corresponding to Beauville surfaces, and the extension of Theorem 230 to all automorphisms σ different from complex conjugation.…”

An explicit quasiplatonic curve with non-abelian moduli field

“…Although no explicit examples are in the literature, the existence of regular dessins with non-abelian field of moduli follows from a theorem of Jarden [8], and they are known to exist even if one fixes the type of the dessin [4]. We prove that the underlying curve of one of them actually has itself the same field of moduli, so we also provide an example of a quasiplatonic curve with non-abelian field of moduli.…”

“…To prove that this is impossible, we use the results described in [2]. Our particular dessin is of type (6,4,6), so it can be seen as the map C ∼ = Γ\H → ∆(6, 4, 6)\H ∼ = P 1 , for some Γ ∆ (6,4,6), where ∆ (6,4,6) is the triangle group with parameters (6, 4, 6) acting on H in the usual way. The results in [2] ensure that the only way that this dessin can be non-maximal is if ∆(6, 4, 6) is included with finite index in another triangle group ∆ , and by this inclusion Γ < ∆ is a normal subgroup, so that the dessin Γ\H → ∆ \H is regular.…”

“…These dessins d'enfants have monodromy actions corresponding to the following homomorphisms from F 2 to S 12 : The Galois action preserves regular covers, since G Q acts as automorphisms of F 2 (see [4]). Therefore, if for σ ∈ G Q , σ(…”

Bipartite graph embeddings, Riemann surfaces and Galois groups

“…An important recent development has been the proof by González-Diez and Jaikin-Zapirain [28] that G acts faithfully on the most symmetric dessins, namely the regular dessins, those with an automorphism group acting transitively on the edges of the embedded graph. Many explicit examples of such dessins arise from the search by topological graph theorists for the most symmetric surface embeddings of various classes of arc-transitive graphs, starting with the work of Heffter [33] and Biggs [4,5] on complete graphs.…”