Is being a higher rank lattice a first order property?

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.