Adelic superrigidity and profinitely solitary lattices

TL;DR

This paper establishes an adelic superrigidity theorem linking algebraic groups over number fields to profinite commensurability classes of lattices in higher rank Lie groups, with implications for profinite rigidity.

Contribution

It introduces an adelic version of superrigidity that characterizes when algebraic groups are adelically isomorphic based on profinite classes of lattices.

Findings

Proves adelic superrigidity for higher rank lattices.

Shows profinite classes correspond to adelic isomorphism of algebraic groups.

Discusses implications for profinite rigidity questions.

Abstract

By arithmeticity and superrigidity, a commensurability class of lattices in a higher rank Lie group is defined by a unique algebraic group over a unique number subfield of $\mathbb{R}$ or $\mathbb{C}$. We prove an adelic version of superrigidity which implies that two such commensurability classes define the same profinite commensurability class if and only if the algebraic groups are adelically isomorphic. We discuss noteworthy consequences on profinite rigidity questions.