The PI will continue research on lattices in locally compact groups which are automorphism groups of polyhedral complexes. The subject of lattices in Lie groups has a long history, but fundamental questions about lattices in more general topological groups remain unanswered. The research proposed addresses these basic questions using a variety of techniques: combinatorial, geometric, algebraic and analytical. This breadth, and the classical nature of the questions being asked, means that the proposal connects several mathematical subfields, in particular geometric group theory and the topics of lattices, rigidity, buildings, and algebraic and Kac-Moody groups. In analogy with Bass-Serre theory, used for studying tree lattices, the theory of complexes of groups and their coverings is used as a powerful tool for studying lattices in automorphism groups of higher-dimensional complexes.
Lattices arise as discrete sets of symmetries of many spaces, and are important in physics and chemistry. The PI is interested in lattices which consist of symmetries of polyhedral complexes, that is, spaces constructed by gluing together polyhedra. The project addresses basic questions about these lattices, and will hopefully give new insights into classical cases. This research also connects several areas of mathematics, in particular geometric group theory and the topics of buildings, linear algebraic groups, and Kac-Moody theory.
