Bi-interpretability with $\mathbb{Z}$ and models of the complete elementary theories of ${\text{SL}}_n(\mathcal{O})$, ${\text{T}}_n(\mathcal{O})$ and ${\text{GL}}_n(\mathcal{O})$, $n\ge 3$

Abstract

Let $\mathcal{O}$ be the ring of integers of a number field, and let $n\ge 3$. This paper studies bi-interpretability of the ring of integers $\mathbb{Z}$ with the special linear group ${\text{SL}}_n(\mathcal{O})$, the general linear group ${\text{GL}}_n(\mathcal{O})$ and solvable group of all invertible uppertriangular matrices over $\mathcal{O}$, ${\text{T}}_n(\mathcal{O})$. For each of these groups we provide a complete characterization of arbitrary models of their complete elementary theories.

arXiv comment19 pages. arXiv admin note: text overlap with arXiv:1609.09802

References in corpus (1)

• Distinguishing every finitely generated field of characteristic \neq2 by a single field axiom

Cited by in corpus (3)

• On the model theory of higher rank arithmetic groups
• On first order rigidity for linear groups
• Elementary equivalence of Kac-Moody groups