  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.
