The proposal addresses various asymptotic questions in the realm of residually finite groups. The core object of the proposal is a descending chain of finite index subgroups with trivial intersection. These chains give rise to actions on locally finite rooted trees by automorphisms. There is an interesting interplay between the ergodic action on the boundary of such a tree and the finite actions of the coset spaces of the subgroups. One may assign various natural invariants to these chains and actions that turn out to relate to well-known notions in group theory, like amenability, Betti numbers, expanders, bounded generation, the cost, largeness or self-similarity. The methods used in the project borrow tools from topology, probability, number theory, ergodic theory and analysis. Besides group theory, solving the proposed problems would have impact on ergodic theory, number theory and topology.
The set of symmetries of any structure forms a group. Thus group theory naturally comes into the picture whenever one needs to analyze symmetries (for instance in chemistry or quantum physics, or inside mathematics, in topology, geometry, number theory and the theory of codes). The main goal of the proposed project is to understand naturally appearing mathematical objects in terms of various estimates on the groups attached to them. Besides establishing surprising connections between distant mathematical principles and thus leading to deep and useful results, the project also has elementary aspects that can be presented on an undergraduate level and thus serve educational purposes. Part of the proposal is to work out and start a new high level Inquiry Based course in algebra, leading to the publication of experimental teaching material.
