The Cohomological Sarnak-Xue Density Hypothesis for $S{O}_5$

We prove the cohomological version of the Sarnak–Xue Density Hypothesis for $S{O}_5$ over a totally real field and for inner forms split at all finite places. The proof relies on recent lines of work in the Langlands program: (i) Arthur’s Endoscopic Classification of Representations of classical groups, extended to inner forms by Taïbi and its explicit description for $S{O}_5$ by Schmidt, and (ii) the Generalized Ramanujan–Petersson Theorem, proved for cohomological self-dual cuspidal representations of general linear groups. We give applications to the growth of cohomology of arithmetic manifolds, density-Ramanujan complexes, cutoff phenomena and optimal strong approximation.