Profinite properties of lattices in Lie groups

Which invariants of a group are profinite in the sense that they are already determined by the set of the finite quotient groups? This question has gained a lot of attention in recent years and it is particularly intriguing for discrete subgroups of semisimple Lie groups with cofinite Haar measure. In this talk, we will report on recent progress in the field with a special view towards L²-cohomological invariants. We will see that some of these invariants are profinite while others are not and we will draw geometric conclusions on the Euler characteristic of the associated locally symmetric spaces.