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 boolean AND(n is an element of N) {g(n) : g is an element of G*}. We construct an expansion of Z by a predicate (Z, +, P) such that the type-connected component Z(empty set)*(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.