The Cohomological Sarnak-Xue Density Hypothesis for $SO_5$

TL;DR

This paper proves a cohomological version of the Sarnak--Xue Density Hypothesis for the group SO_5 over totally real fields, utilizing recent advances in the Langlands program and applications to arithmetic and geometric properties.

Contribution

It establishes the cohomological Sarnak--Xue Density Hypothesis for SO_5, extending previous work through new applications of Arthur's classification and Ramanujan--Petersson results.

Findings

Proved the density hypothesis for SO_5 over totally real fields.

Derived bounds on cohomology growth of arithmetic manifolds.

Applied results to density-Ramanujan complexes and approximation phenomena.

Abstract

We prove the cohomological version of the Sarnak--Xue Density Hypothesis for $SO_{5}$ 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 Ta\"ibi and its explicit description for $SO_{5}$ 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.