On the profinite rigidity of lattices in higher rank Lie groups

Abstract

We investigate which higher rank simple Lie groups admit profinitely but not abstractly commensurable lattices. We show that no such examples exist for the complex forms of type $E_8$, $F_4$, and $G_2$. In contrast, there are arbitrarily many such examples in all other higher rank Lie groups, except possibly $\mathrm{SL}_{2n+1}(\mathbb{R})$, $\mathrm{SL}_{2n+1}(\mathbb{C})$, $\mathrm{SL}_n(\mathbb{H})$, or groups of type $E_6$.