Higher Kazhdan property and unitary cohomology of arithmetic groups

We prove that the restriction in unitary cohomology from the real points of a simple algebraic group $\mathbf{G}$ over a number field $k$ to the arithmetic group is an isomorphism in degrees below the rank. Unitary cohomology means (continuous) cohomology with arbitrary unitary coefficients. For the adelic version we obtain an isomorphism in all degrees. Results about the bijectivity of the restriction map are related to vanishing results for the cohomology of $\mathbf{G}(k\otimes\mathbb{R})$ and $\mathbf{G}(\mathscr{O})$ with unitary coefficients. A prototype example of our vanishing results is that the cohomology of $SL_n(\mathbb{Z})$ vanishes for all unitary coefficients without invariant vectors and all degrees below $n-1$. We regard this phenomenon as a higher form of Kazhdan property T. We put an emphasis on arbitrary unitary coefficients. Previous works often were limited to irreducible representations or unitary inductions of finite-dimensional representations. A novelty of our approach is the use of methods from geometric group theory and -- in the rank 1 case -- from (global) representation theory pertaining to the spectral gap property.