First order rigidity of non-uniform higher rank arithmetic lattices

TL;DR

This paper proves that any finitely generated group elementarily equivalent to a non-uniform higher-rank arithmetic lattice is actually isomorphic to it, establishing a strong rigidity property in model theory.

Contribution

It demonstrates first-order rigidity of non-uniform higher-rank arithmetic lattices, showing they are uniquely characterized by their elementary theory.

Findings

Any finitely generated group elementarily equivalent to such a lattice is isomorphic to it.

Establishes a form of model-theoretic rigidity for these lattices.

Extends rigidity results to the realm of first-order logic for higher-rank arithmetic groups.

Abstract

If $\Gamma$ is an irreducible non-uniform higher-rank characteristic zero arithmetic lattice (for example, $SL_n(\mathbb{Z})$, $n \geq 3$) and $\Lambda$ is a finitely generated group that is elementarily equivalent to $\Gamma$, then $\Lambda$ is isomorphic to $\Gamma$.