  We show that the Galois group $Gal(\bar{\Q} /\Q)$ operates faithfully on the
set of connected components of the moduli spaces of surfaces of general type,
and also that for each element $\sigma \in Gal(\bar{\Q} /\Q)$ different from
the identity and from complex conjugation, there is a surface of general type
such that $X$ and the Galois conjugate variety $X^{\sigma}$ have nonisomorphic
fundamental groups. The result was announced by the second author at the
Alghero Conference 'Topology of algebraic varieties' in september 2006. Before
the present paper was actually written, we received a very interesting preprint
by Robert Easton and Ravi Vakil (\cite{e-v}), where it is proven, with a
completely different type of examples, that the Galois group $Gal(\bar{\Q}
/\Q)$ operates faithfully on the set of irreducible components of the moduli
spaces of surfaces of general type. We also give other simpler examples of
surfaces with nonisomorphic fundamental groups which are Galois conjugate,
hence have isomorphic algebraic fundamental groups.
