Betti numbers of arithmetic locally symmetric spaces

The cohomology of arithmetic groups, or equivalently of the associated locally symmetric spaces, is an object of interest in number theory, topology and group theory. Its behaviour is constrained quite strongly by the volume; in this talk I will present a very general result about this obtained in joint work with Mikolaj Fraczyk and Sebastian Hurtado. The main tools are a non-effective result using Benjamini--Schramm convergence, the fine description of arithmetic lattices, counting estimates and sharp upper bounds for orbital integrals; most of the talk will be used to describe what these are and how they fit together to prove the result.