Homotopy type and homology versus volume for arithmetic locally symmetric spaces

We study locally symmetric spaces associated with arithmetic lattices in semisimple Lie groups. We prove the following results about their topology: the minimal number of tetrahedra needed for a triangulation is at most linear in the volume and the Betti numbers are sub-linear in the volume except possibly in middle degree. The proof of these results uses the geometry of these spaces, namely the study of their thin parts. In this regard we prove that these spaces converge in the Benjamini--Schramm sense to their universal covers and give an explicit bound for the volume of the thin part for trace fields of large degree. The main technical ingredients for our proofs are new estimates on orbital integrals, a counting result for elements of small displacement, and a refined version of the Margulis lemma for arithmetic locally symmetric spaces.
Comment: 58 pages