Abert will investigate the asymptotic behavior of natural invariants on the subgroup lattice of residually finite groups. Examples for such invariants are rank, cost, Betti numbers, Heegaard genus, amenability, spectral gap, bounded generation and girth. The historical background of the project is subgroup growth and profinite groups (Lubotzky, Segal, Shalev and Wilson), finitely presented groups and topology (Luck and Lackenby), orbit equivalence (Weiss, Popa and Furman) and self-similar groups (Grigorchuk and Nekrasevich). A 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, and specifically, arithmetic groups. The core object of interest is a descending chain of finite index subgroups. There is an interesting interplay between the combinatorics of the finite coset actions in the chain, the dynamics of the measure preserving action on the boundary of the corresponding coset tree and the structure of the discrete group. This interplay allows one e.g. to apply rigidity theorems in measurable group theory and get new graph theoretical results. The proposed activity also has connections to percolation on transitive graphs and the theory of 3-manifolds.
Group theory is an old and central mathematical principle, born in the early 19th century. The set of symmetries of an arbitrary object forms a group, so groups arise virtually in all areas in mathematics and also in certain parts of physics and chemistry. Abert will study residually finite groups; these are natural meeting points of finite and infinite groups. The proposed activity lies at the crossroads of group theory, graph theory and dynamics and has strong connections to certain areas in probability theory and topology; as such, it is highly interdisciplinary. As part of the project, Abert will work with gifted undergraduate and graduate students and expose them to parts of his research through creative problem solving. The project is also coordinated with the University of Chicago VIGRE Program. The ultimate goal is to work out a core format for inquiry based learning on research level.
