Model theoretic connected components of finitely generated nilpotent groups

We prove that for a finitely generated infinite nilpotent group G with structure (G, . , . . . ), the connected component G*⁰ of a sufficiently saturated extension G* of G exists and equals $\bigcap\limits_{n \in \mathbb{N}} {\left\{ {{g^n}:g \in {G^*}} \right\}} $ We construct an expansion of ℤ by a predicate (ℤ, +, P) such that the type-connected component ${\mathbb{Z}^*}_\emptyset ^{00}$ 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 for finite partitions of groups.