Higher property T and below-rank phenomena of lattices

Abstract

The purpose of this paper is twofold. We explore higher property T as an abstract group-theoretic property. In particular, we provide new operator-algebraic characterizations of higher property T. Then we turn to lattices in semisimple Lie groups. We relate higher property T to other cohomological, rigidity and geometric phenomena below the real rank. The second part outlines a conjectural framework that unifies these aspects and reviews recent advances.

45 pp; an abridged version will appear in the Proceedings of the ICM 2026; added new applications and clarified relations between Banach und unitary results; added alternative proof of Ozawa's criterion and additional references