Some model theory of the Heisenberg group

TL;DR

This paper establishes a deep connection between the model completeness of a field and its associated Heisenberg group, extending automorphism results and analyzing logical properties like quantifier elimination.

Contribution

It proves that a field's model completeness is equivalent to that of its Heisenberg group and extends automorphism results to monomorphisms, also discussing interpretability issues.

Findings

Field $K$ is model complete iff $H(K)$ is model complete.

$H(K)$ lacks quantifier elimination.

$H(K)$ is not bi-interpretable with $K$.

Abstract

We show that a field $K$ is model complete (in the language of rings) if and only if the Heisenberg group $H(K)$ is model complete (in the language of groups). To show that, we extend Levchuk's result about automorphisms of $H(K)$ to the case of monomorphisms $H(K)\to H(M)$. We also show that $H(K)$ does not have quantifier elimination and discuss its (non-)bi-interpretability with $K$.