On absolutely profinitely solitary lattices in higher rank Lie groups

TL;DR

This paper investigates conditions under which certain noncocompact lattices in higher rank Lie groups are uniquely determined by their profinite completions, linking group properties to conjectures in Grothendieck rigidity.

Contribution

It establishes new criteria for absolute profinite solitariness of specific lattices in higher rank Lie groups, contingent on a conjecture in Grothendieck rigidity.

Findings

Noncocompact lattices in specified Lie groups are absolutely solitary if a Grothendieck rigidity conjecture holds.

Cocompact lattices generally are not absolutely solitary.

Profinite completion can determine the commensurability class of certain lattices.

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.