Geometric, Topological, and Dynamical Problems in Group Theory

This proposal is primarily in the area of geometric group theory, which connects two foundational fields of mathematics, namely group theory and geometry/topology. A group can be thought of as the set of symmetries of an object or a space such as a water molecule or a fractal. The geometric properties of the space can often reveal algebraic properties of the group. This project aims to explore these connections between groups and the spaces on which they act. Broader impacts of the proposal include mentoring students and organizing conferences and special sessions at meetings. The focus of this project is on groups which arise as symmetry groups of Cantor sets. Cantor sets have the unique property that, when divided in half, give two copies of the original Cantor set. Many examples and counterexamples in group theory arise as symmetry groups of Cantor sets, including Grigorchuk's group and the family of Thompson's groups. Although these groups have been around for many years, they are still driving examples in modern group theory. The first goal of the project is to better understand Thompson's groups themselves via the Poisson boundary of F and the maximal subgroups of V. The second goal is to use related constructions to gain a better understanding of groups having some form, or acting on a space with some form, of nonpositive curvature. The final goal is to obtain applications in model theory and dynamics. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.