Higher Kazhdan property and unitary cohomology of arithmetic groups

TL;DR

This paper extends Garland's theorem to simple Lie groups and their lattices, linking higher Kazhdan properties with unitary cohomology vanishing, and improves stability results for cohomology invariance in semisimple Lie groups.

Contribution

It generalizes Garland's theorem to a broader class of groups and lattices, and enhances the understanding of cohomology invariance with arbitrary unitary coefficients.

Findings

Proved a version of Garland's theorem for simple Lie groups and lattices.

Improved stability range for cohomology invariance.

Applied geometric group theory and spectral gap methods in the proof.

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.