#### Vol. 313, No. 1, 2021

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Adelic superrigidity and profinitely solitary lattices

### Holger Kammeyer and Steffen Kionke

Vol. 313 (2021), No. 1, 137–158
DOI: 10.2140/pjm.2021.313.137
##### 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 $ℝ$ or $ℂ$. 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.

##### Keywords
profinite rigidity, arithmeticity, superrigidity
##### Mathematical Subject Classification
Primary: 20E18, 22E40