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

TL;DR

This paper characterizes models of the complete elementary theories of certain matrix groups over the ring of integers of a number field, establishing bi-interpretability with the ring of integers for these groups.

Contribution

It provides a complete characterization of models of the elementary theories of $ ext{SL}_n( ext{O})$, $ ext{T}_n( ext{O})$, and $ ext{GL}_n( ext{O})$ for $n \\geq 3$, demonstrating bi-interpretability with $\\mathbb{Z}$.

Findings

Models of the theories are fully characterized.

Bi-interpretability with the ring of integers is established.

Results apply to groups over number fields for $n \\geq 3$.

Abstract

Let $\mathcal{O}$ be the ring of integers of a number field, and let $n\geq 3$. 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.