ON THE MODEL THEORY OF HIGHER RANK ARITHMETIC GROUPS

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