Submission date: 14 December 2011.

Abstract:

We prove that for a finitely generated infinite nilpotent group G with a first order structure (G,*,...), the connected component G*0 of a sufficiently saturated extension G* of G exists and equals \bigcap_{n\in\N} {g^n : g\in G^*}. We construct a first order expansion of Z by a predicate (Z,+,P) such that the type-connected component Z*00_{\emptyset} 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.

Mathematics Subject Classification: 03C60, 20F16, 05E15, 20A15

Keywords and phrases:

Full text arXiv 1112.3292: pdf, ps.