  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.
