Model theoretic connected components of finitely generated nilpotent groups

We prove that for a finitely generated infinite nilpotent group G with a first order structure (G, ·, . . .), the connected component G of a sufficiently saturated extension G of G exists and equals ⋂ n∈N {g : g ∈ G}. We construct a first order expansion of Z by a predicate (Z,+, P ) such that the type-connected component Z ∅ is strictly smaller than Z. 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.