Galois cohomology and profinitely solitary Chevalley groups

TL;DR

This paper investigates whether the profinite completion of certain arithmetic groups uniquely determines their commensurability class, using Galois cohomology and assuming Grothendieck rigidity.

Contribution

It provides a Galois cohomological approach to establish conditions under which Chevalley groups are profinitely solitary, advancing understanding of their algebraic and arithmetic properties.

Findings

Proves profinite solitariness for certain Chevalley groups under Grothendieck rigidity

Connects Galois cohomology with the classification of arithmetic groups

Offers a framework for identifying groups from their profinite completions

Abstract

For every number field and every Cartan Killing type, there is an associated split simple algebraic group. We examine whether the corresponding arithmetic subgroups are profinitely solitary so that the commensurability class of the profinite completion determines the commensurability class of the group among finitely generated residually finite groups. Assuming Grothendieck rigidity, we essentially solve the problem by Galois cohomological means.