On the model theory of higher rank arithmetic groups

Abstract

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