  We take the fundamental group of the complement of the branch curve of a
generic projection induced from canonical embedding of a surface. This group is
stable on connected components of moduli spaces of surfaces. Since for many
classes of surfaces it is expected that the fundamental group has a polycyclic
structure, we define a new invariant that comes from this structure. We compute
this invariant for a few examples. Braid monodromy factorizations related to
curves is a first step in computing the fundamental group of the complement of
the curve, and thus we also indicate the possibility of using braid monodromy
factorizations of branch curves as an invariant of a surface.
