Group theory is a central principle in mathematics. The set of symmetries of an arbitrary mathematical object forms a group, so groups arise virtually in all areas in mathematics (and also in certain parts of physics and chemistry). An infinite group is called residually finite, if the intersection of its subgroups of finite index is trivial. This means that finite images approximate the group structure. Important examples are finitely generated linear groups, specifically, arithmetic groups. There are various group invariants, whose asymptotic behavior on the subgroup lattice of such a group is important to understand. Besides pure group theory, questions of this type emerge naturally in algebraic topology, number theory, geometry and representation theory. Examples for these invariants include the rank, homologies and various geometric and spectral invariants of the finite quotients. Miklos Abert, the researcher of the proposal, is an expert in this area. His recent work connects seemingly far areas, like graph theory, 3-manifold theory and topological dynamics through profinite actions. His earlier work analyzes random profinite actions. He proposes to continue his research in these directions and also to engage in emerging new directions, like graph limits. Ultimately, Abert aims to build a general theory of residually finite groups acting on rooted trees. Abert currently holds a tenure track position at the University of Chicago, one of the top ranking universities in the US. He continuously receives individual NSF research grants since 2004. If funded, he intends to return to Europe and continue his research in the Renyi Institute. This would enrich the mathematical culture of Hungary, one of the new Member States to the European Union and would contribute towards reversing brain drain. The Institute has expressed its intention that the researcher joins it permanently in case the project is successfully completed.