Growth of Cohomology of Arithmetic Groups and the Stable Trace Formula

We study growth of Betti numbers in towers of cocompact arithmetic lattices in unitary groups U(a,b). In the middle degree of cohomology, the Betti numbers grow proportionally to the volume of the manifold, but away from the middle degree, the growth is known to be sub-linear in the volume. After rephrasing the problem into representation-theoretic terms, we give upper bounds on the growth of cohomology in small degrees coming from certain families of representations. These upper bounds are achieved in the framework of the endoscopic classification of representations: we use Arthur's stable trace formula to bound the growth in terms of multiplicities of discrete series representations on endoscopic groups