$L^{2}$–invariants of nonuniform lattices in semisimple Lie groups

Abstract

We compute $L^2$–invariants of certain nonuniform lattices in semisimple Lie groups by means of the Borel–Serre compactification of arithmetically defined locally symmetric spaces. The main results give new estimates for Novikov–Shubin numbers and vanishing $L^2$–torsion for lattices in groups with even deficiency. We discuss applications to Gromov’s zero-in-the-spectrum conjecture as well as to a proportionality conjecture for the $L^2$–torsion of measure-equivalent groups.