On absolutely profinitely solitary lattices in higher rank Lie groups

Abstract

We establish conditions under which lattices in certain simple Lie groups are profinitely solitary in the absolute sense, so that the commensurability class of the profinite completion determines the commensurability class of the group among finitely generated residually finite groups. While cocompact lattices are typically not absolutely solitary, we show that noncocompact lattices in $\operatorname{Sp}(n,\mathbb{R})$, $G_{2(2)}$, $E_8(\mathbb{C})$, $F_4(\mathbb{C})$, and $G_2(\mathbb{C})$ are absolutely solitary if a well-known conjecture on Grothendieck rigidity is true.