Is being a higher rank lattice a first order property?

TL;DR

This paper demonstrates that the property of being a higher rank lattice can be characterized by a specific first-order sentence in the language of groups, linking algebraic structure to logical definability.

Contribution

It introduces a first-order sentence that precisely characterizes groups isomorphic to higher rank lattices of the form PSL_n(O), with n ≥ 3 and O a ring of S-integers.

Findings

A first-order sentence characterizes higher rank lattices.

Higher rank lattice property is first-order definable.

Connection between algebraic groups and logical properties.

Abstract

We show that there is a sentence $\varphi$ in the first order language of groups such that a finitely generated group $\Gamma$ satisfies $\varphi$ if and only if $\Gamma$ is isomorphic to a group of the form $\mathrm{PSL}_n(O)$, where $n \geq 3$ and $O$ is a ring of $S$-integers in a number field.