  The paper contains a general construction which produces new examples of non
simply-connected smooth projective surfaces. We analyze the resulting surfaces
and their fundamental groups. Many of these fundamental groups are expected to
be non-residually finite. Using the construction we also suggest a series of
potential counterexamples to the Shafarevich conjecture which claims that the
universal covering of smooth projective variety is holomorphically convex. The
examples are only potential since they depend on group theoretic questions,
which we formulate, but we do not know how to answer. At the end we formulate
an arithmetic version of the Shafarevich conjecture.
