Higher Kazhdan property and unitary cohomology of arithmetic groups
Uri Bader, Roman Sauer
TL;DR
The paper extends Garland's higher property T to real semisimple Lie groups and their lattices, establishing vanishing of unitary cohomology in degrees up to a rank-determined bound with arbitrary unitary coefficients. It introduces polynomial cohomology and leverages geometric group theory, Shapiro-type inductions, and spectral-gap techniques (including Clozel-type results) to relate the cohomology of lattices, adelic groups, and ambient semisimple groups. The authors prove stable cohomology isomorphisms in low degrees across lattices and their ambient groups, derive a product-formula for the cohomological invariants $r_0$ and $r$, and give adelic generalizations that connect $k$-points, adeles, and automorphic representations. These results yield higher-T-type vanishing, refined stability ranges, and a clearer understanding of the cohomological unitary dual, with applications to stability, invariants under measure equivalence, and arithmeticity phenomena. The work thus provides a comprehensive framework for higher Kazhdan properties in arithmetic groups, combining representation theory, cohomology, and geometric-group-theoretic methods to yield broad, structural insights.
Abstract
Notions of higher Kazhdan property can be defined in terms of vanishing of unitary group cohomology in higher degrees. Garland's theorem for simple groups over non-archimedean fields provides the first examples of a higher Kazhdan property. We prove a version of Garland's theorem for simple Lie groups and their lattices. We generalize theorems of Borel and Borel-Yang about the invariance of the cohomology of lattices in semisimple Lie groups and adelic groups by improving the stability range and allowing for arbitrary unitary representations as coefficients. A novelty of our approach is the use of methods from geometric group theory and -- in the case of rank 1 -- from Clozel's work on the spectral gap property.