Faithful actions of the absolute Galois group on connected components of moduli spaces

Abstract

We use a canonical procedure associating to an algebraic number $a$ first a hyperelliptic curve $C_a$, and then a triangle curve $(D_a, G_a)$ obtained through the normal closure of an associated Belyi function. In this way we show that the absolute Galois group ${{\mathrm{Gal}}}(\bar{{\mathbb {Q}}} /{\mathbb {Q}})$ acts faithfully on the set of isomorphism classes of marked triangle curves, and on the set of connected components of marked moduli spaces of surfaces isogenous to a higher product (these are the free quotients of a product $C_1 \times C_2$ of curves of respective genera $g_1, g_2 \ge 2$ by the action of a finite group $G$). We show then, using again the surfaces isogenous to a product, first that it acts faithfully on the set of connected components of the moduli space of surfaces of general type (amending an incorrect proof in a previous arXiv version of the paper); and then, as a consequence, we obtain our main result: for each element $\sigma \in {{\mathrm{Gal}}}(\bar{{\mathbb {Q}}} /{\mathbb {Q}})$, not in the conjugacy class of complex conjugation, there exists a surface of general type $X$ such that $X$ and the Galois conjugate surface $X^{\sigma }$ have nonisomorphic fundamental groups. Using polynomials with only two critical values, we can moreover exhibit infinitely many explicit examples of such a situation.