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.

Given any sequence of non-abelian finite simple primitive permutation groups $(S_{n})$, we construct a finitely generated group $G$ whose profinite completion is the infinite permutational wreath product $\ldots S_{n}\wr S_{n-1}\wr\ldots\wr S_{0}$. It follows that the upper composition factors of $G$ are exactly the groups $S_{n}$. By suitably choosing the sequence $(S_{n})$ we can arrange that $G$ has any one of a continuous range of slow, non-polynomial subgroup growth types. We also construct a $61$-generator perfect group that has every non-abelian finite simple group as a quotient. 2000 Mathematics Subject Classification: 20E07, 20E08, 20E18, 20E32.