Adelic superrigidity and profinitely solitary lattices

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