On the model theory of higher rank arithmetic groups

Abstract

Let $\Gamma$ be a centerless irreducible higher rank arithmetic lattice in characteristic zero. We prove that if $\Gamma$ is either non-uniform or is uniform of orthogonal type and dimension at least 9, then $\Gamma$ is bi-interpretable with the ring $\mathbb{Z}$ of integers. It follows that the first order theory of $\Gamma$ is undecidable, that all finitely generated subgroups of $\Gamma$ are definable, and that $\Gamma$ is characterized by a single first order sentence among all finitely generated groups.

preliminary version. Comments welcome

References in corpus (1)

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