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 We construct an expansion of ℤ by a predicate (ℤ, +, P ) such that the type-connected component 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.