Some model theory of the Heisenberg group

TL;DR

This work establishes a precise equivalence between model completeness of a field $K$ and its Heisenberg group $H(K)$ when viewed as a group, showing that $H(K)$ is model complete if and only if $K$ is model complete in the language of rings. The authors extend Levchuk's automorphism classification to monomorphisms $H(K)\to H(M)$ by analyzing central group extensions and applying Maltsev's interpretation of $K$ in $H(K)$, enabling transfer of model-theoretic properties via interpretability functors. They further show that $H(K)$ does not admit quantifier elimination and study (non-)bi-interpretability with $K$, obtaining negative results for several natural classes of fields. The results illuminate the model-theoretic behavior of unipotent algebraic groups and lay groundwork for extending the approach to broader classes of unitriangular groups.

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 $K$.

Theorems & Definitions (27)

• Remark 2.1
• Lemma 2.2
• Lemma 2.3
• proof
• Lemma 2.4
• proof
• Theorem 2.5
• proof
• proof : Proof of Claim
• Theorem 2.6: Levchuk