Congruence rigidity of algebraic groups

TL;DR

This paper characterizes certain algebraic groups over number fields by their finite adele points, extending the concept of profinite rigidity to a broader class of groups under specific assumptions.

Contribution

It generalizes previous results on split groups to arbitrary groups, providing a broader understanding of profinite rigidity in algebraic groups.

Findings

Identifies simple algebraic groups over number fields determined by finite adele points

Shows that higher rank arithmetic groups are profinitely solitary under CSP and Grothendieck rigidity

Extends previous work from split groups to all algebraic groups

Abstract

We identify the simple algebraic groups over number fields that are, in a suitable sense, determined by their finite adele points. Assuming CSP and Grothendieck rigidity, our results essentially characterize higher rank arithmetic groups that are profinitely solitary: the profinite commensurability class determines the commensurability class among finitely generated residually finite groups. This generalizes previous work of the second author with R. Spitler from split groups to arbitrary groups.