The long-term objective of the PI is to understand groups considered as geometric objects, the geometry and topology of spaces modeling these groups, and topological aspects of the group actions on such spaces. Specific objectives include the analysis and determination of higher dimensional filling invariants of groups, the establishment of foundational results for these invariants, and development of an approach to the relation gap problem via combinatorial Morse theory. The PI also proposes to study promotion maps and their properties, as well as applications to open problems.
Group theory is the theory of symmetry, which plays a fundamental role in mathematics. Patterns of symmetries (or "groups") arise very naturally in geometry, but also in other, more abstract, fields of mathematics. In geometric group theory one studies abstract groups by constructing geometric spaces, tailor-made for the groups at hand, in order to "see" the symmetry and better understand it. This project involves the detailed study of the geometric spaces thus constructed. In particular, it concerns the determination of shape and related properties of geometric spaces based on the nature of the symmetries they possess.