  We prove that for a finitely generated infinite nilpotent group G with a
first order structure (G,*,...), the connected component G*0 of a sufficiently
saturated extension G* of G exists and equals $\bigcap_{n\in\N} {g^n : g\in
G^*}$. We construct a first order expansion of Z by a predicate (Z,+,P) such
that the type-connected component Z*00_{\emptyset} is strictly smaller than
Z*0. We generalize this to finitely generated virtually solvable groups. As a
corollary of our construction we obtain an optimality result for the van der
Waerden theorem.
