  We give infinite series of groups Gamma and of compact complex surfaces of
general type S with fundamental group Gamma such that 1) Any surface S' with
the same Euler number as S, and fundamental group Gamma, is diffeomorphic to S.
2) The moduli space of S consists of exactly two connected components,
exchanged by complex conjugation.
  Whence, i) On the one hand we give simple counterexamples to the DEF = DIFF
question whether deformation type and diffeomorphism type coincide for
algebraic surfaces. ii) On the other hand we get examples of moduli spaces
without real points. iii) Another interesting corollary is the existence of
complex surfaces S whose fundamental group Gamma cannot be the fundamental
group of a real surface. Our surfaces are surfaces isogenous to a product;
i.e., they are quotients (C_1 X C_2)/G of a product of curves by the free
action of a finite group G. They resemble the classical hyperelliptic surfaces,
in that G operates freely on C_1, while the second curve is a triangle curve,
meaning that $C_2 / G \equiv \PP ^1 and the covering is branched in exactly
three points.
