Description
$L^2$-invariants are defined for the universal covering of a compact manifold as counterparts of classical invariants such as Betti numbers, the index of elliptic operators, or the analytic torsion. Their definition uses the group von Neumann algebra. They have important applications in topology and geometry. Of particular interest is the question if $L^2$-invariants can be approximated by the corresponding classical invariants for a sequence of coverings of finite degree converging in the Benjamini-Schramm sense to the universal covering. In this talk I will discuss this problem for locally symmetric spaces of finite volume and the analytic torsion. The basic tool is the Arthur-Selberg trace formula. If time permits I will also discuss some applications to the cohomology of arithmetic groups.