Model theoretic connected components of finitely generated nilpotent groups

TL;DR

This paper investigates the model-theoretic connected components of finitely generated nilpotent groups, establishing their structure in saturated extensions and providing counterexamples in expanded structures, with applications to combinatorial theorems.

Contribution

It proves the existence and explicit description of connected components in finitely generated nilpotent groups and constructs a counterexample in an expanded structure, extending to virtually solvable groups.

Findings

Connected component G*0 equals the intersection of all nth powers in G*

Constructed an expansion of Z where the type-connected component is strictly smaller

Derived an optimality result for the van der Waerden theorem

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.