  We give 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 Gal(\bar{\Q} /\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 x C_2 of curves of respective
genera g_1, g_2 >= 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 moduli spaces of surfaces of general type
(amending an incorrect proof in a previous ArXiv version of the paper); and
then, as a consequence, we obtain that for every element \sigma \in
\Gal(\bar{\Q} /\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.
