On the model theory of higher rank arithmetic groups

TL;DR

This paper proves that certain higher rank arithmetic groups are bi-interpretable with the integers, leading to their first order theories being undecidable and their subgroups definable, revealing deep model-theoretic properties.

Contribution

It establishes bi-interpretability of specific higher rank arithmetic groups with the integers, connecting group theory with model theory in a novel way.

Findings

Groups are bi-interpretable with 

First order theories are undecidable

Finitely generated subgroups are definable

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.