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Model theoretic connected components of finitely generated nilpotent groups

Published online by Cambridge University Press:  12 March 2014

Nathan Bowler ,
Cong Chen  and
Jakub Gismatullin
Nathan Bowler
Affiliation:
Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, United Kingdom.E-mail:sfwc@hotmail.com
Cong Chen
Affiliation:
School of Mathematics, University of Leeds, Woodhouse Lane, Leeds, LS2 9JT, UK, E-mail: mmcc@leeds.ac.uk
Jakub Gismatullin
Affiliation:
School of Mathematics, University of Leeds, Woodhouse Lane, Leeds, LS2 9JT, UK Instytut Matematyczny Uniwersytetu Wrocławskiego, Pl. Grunwaldzki 2/4, 50-384 Wrocław, Poland, E-mail: gismat@math.uni.wroc.pl URL: www.math.uni.wroc.pl/~gismat
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Abstract

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.

Keywords

model theoretic connected componentsfinitely generated groupthick setvan der Waerden theorem
Type
Research Article
Information
The Journal of Symbolic Logic , Volume 78 , Issue 1 , March 2013 , pp. 245 - 259
Copyright
Copyright © Association for Symbolic Logic 2013

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References

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