Model theoretic connected components of finitely generated nilpotent groups

Nathan Bowler; Cong Chen; Jakub Gismatullin

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arXiv · 1112.3292

Model theoretic connected components of finitely generated nilpotent groups

Nathan Bowler·Cong Chen·Jakub Gismatullin

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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.

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BibTeX RIS

Nathan Bowler, Cong Chen, Jakub Gismatullin. 2012-09-04. Model theoretic connected components of finitely generated nilpotent groups. https://arxiv.org/abs/1112.3292

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