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

Milestones

Received: 19 February 2021

Revised: 21 June 2021

Accepted: 22 June 2021

Published: 17 September 2021