Although this project is about geometric group theory, the inspiration for many of the methods and questions comes from geometric topology, particularly the geometry and topology of three-manifolds. The main part of the project is on relatively hyperbolic groups. In collaborations with several mathematicians, the PI will apply diverse methods (including group theoretic "Dehn filling") to the understanding of relatively hyperbolic and hyperbolic groups. The goal of these explorations is to shed light on some fundamental problems in hyperbolic groups, especially the question of which hyperbolic groups contain surface subgroups. This question is closely related to the virtual Haken conjecture of Waldhausen. A second part of the project is aimed at developing a general theory of the "variety" of actions of a fixed group on negatively curved (but not necessarily proper) spaces.
Geometric group theory is the study of infinite groups (objects from abstract algebra) using the techniques of geometry and topology. A central idea is that the best way to understand an abstract group is to see it concretely as a group of symmetries of some geometric object, such as a crystal lattice or a rooted tree. While this idea allows the transfer of techniques from diverse areas of mathematics (including theoretical computer science, analysis, geometry, topology, and dynamics) to group theory, it also has led to new insight into these other fields.
