On the profinite rigidity of lattices in higher rank Lie groups

TL;DR

This paper explores the profinite rigidity of lattices in higher rank Lie groups, showing that certain complex forms do not admit profinitely but not abstractly commensurable lattices, while others do, with some exceptions.

Contribution

It identifies which higher rank simple Lie groups have lattices that are profinitely rigid and which do not, clarifying the landscape of lattice rigidity in these groups.

Findings

No profinitely but not abstractly commensurable lattices in complex forms of $E_8$, $F_4$, and $G_2$.

Existence of arbitrarily many such lattices in other higher rank Lie groups.

Possible exceptions include certain forms of $ ext{SL}_{2n+1}( ext{R})$, $ ext{SL}_{2n+1}( ext{C})$, $ ext{SL}_n( ext{H})$, and groups of type $E_6$.

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 $E_6$.