Elementary coordinatization of finitely generated nilpotent groups

Abstract

This paper has two main parts. In the first part we develop an elementary coordinatization for any nilpotent group $G$ taking exponents in a binomial principal ideal domain (PID) $A$. In case that the additive group $A^{+}$ of $A$ is finitely generated we prove using a classical result of Julia Robinson that one can obtain a central series for $G$ where the action of the ring of integers $\mathbb{Z}$ on the quotients of each of the consecutive terms of the series except for one very specific gap, called the special gap, is interpretable in $G$. Then we use a refinement of this central series to give a criterion for elementary equivalence of finitely generated nilpotent groups in terms of the relationship between group extensions and the second cohomology group.

Cited by in corpus (2)

• On groups elementarily equivalent to a group of triangular matrices $T_n(R)$
• Full rank presentations and nilpotent groups: structure, Diophantine problem, and genericity