How much information on a residually finite group can one recover from its finite quotients? This question has been raised in diverse contexts but it is particularly intriguing for lattices in simple Lie groups. In this research project, we want to investigate to what extend \(\ell^2\)-cohomological properties of groups, in particular lattices, are determined by their profinite completions. The latter will also be investigated as dynamical systems when endowed with translation actions. As geometric outcomes, we will identify situations in which the sign of the Euler characteristic or the volume of a locally symmetric space are determined by the deck transformation groups of its finite sheeted coverings. We will investigate algebraic approximation properties of \(\ell^2\)-cohomology, which should give new insights in the algebraic eigenvalue property. One goal is to establish the Atiyah conjecture for left-orderable groups.
We will start to investigate profinite aspects of \(\ell^2\)-cohomology using rigidity properties of lattices. In the course of this project we further plan to investigate in how far results for lattices extend to larger classes of residually finite groups. For instance, while \(\ell^2\)-acyclicity is a profinite invariant of higher rank lattices, it is unknown whether this holds true for arbitrary residually finite groups. This line of investigations links our project to group theoretic problems which arise in the attempt to construct counterexamples.