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March 2013 Model theoretic connected components of finitely generated nilpotent groups
Nathan Bowler, Cong Chen, Jakub Gismatullin
J. Symbolic Logic 78(1): 245-259 (March 2013). DOI: 10.2178/jsl.7801170

Abstract

We prove that for a finitely generated infinite nilpotent group $G$ with structure $(G,\cdot,\dots)$, the connected component ${G^*}^0$ of a sufficiently saturated extension $G^*$ of $G$ exists and equals \[ \bigcap_{n\in\N} \{g^n\colon g\in G^*\}. \] We construct an expansion of ${\mathbb Z}$ by a predicate $({\mathbb Z},+,P)$ such that the type-connected component ${{\mathbb Z}^*}^{00}_{\emptyset}$ is strictly smaller than ${{\mathbb Z}^*}^0$. 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.

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Nathan Bowler. Cong Chen. Jakub Gismatullin. "Model theoretic connected components of finitely generated nilpotent groups." J. Symbolic Logic 78 (1) 245 - 259, March 2013. https://doi.org/10.2178/jsl.7801170

Information

Published: March 2013
First available in Project Euclid: 23 January 2013

zbMATH: 1272.03145
MathSciNet: MR3087074
Digital Object Identifier: 10.2178/jsl.7801170

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Rights: Copyright © 2013 Association for Symbolic Logic

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Vol.78 • No. 1 • March 2013
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