In this research project the PI proposes to study different problems related to geometry, analysis and probability on discrete groups. The groups the PI is interested in are lattices in Lie groups, groups generated by finite automata, random and hyperbolic groups in the sense of Gromov, groups acting on buildings. The PI found recently new examples of groups with property (T) and the PI intends to continue the study of groups with this property. In particular, the PI would like to find other groups with property (T) and continue the study of automata groups. During this study the PI plans to investigate properties of these groups related to amenability, torsion, growth and the following famous conjectures: the Atiyah conjecture about L2 Betti numbers, the Baum-Connes conjecture, and the Kaplansky-Kadison conjecture about the idempotents. The PI proposes also to continue the study of spectral properties of random walks operators and more generally the Hecke operators for groups generated by finite automata and buildings.
Property (T) is a quite abstract property of some groups, but which found applications in constructions of computer and communication networks. The PI proposes to find new groups with this property which would have applications to several problems related to efficiency of certain algorithms. The other project concerning automata groups combines the problems from the abstract group theory and applications in computer science. The PI plans to give several mini-courses for graduate students based on this research.
