Principal Investigator: Noel Brady
The long-term objective of this research agenda is to understand the large scale geometry of groups and complexes. Specific objectives include the analysis of higher dimensional filling invariants of groups, higher filling invariants at infinity (higher divergence), and investigating the structure of asymptotic cones of finitely presented groups. Other specific objectives include the use of combinatorial Morse theory in understanding virtual fibering in certain classes of groups, and in developing an approach to the relator gap problem. The principal investigator also intends to compare non-positive curvature methods and topological methods (using stable commutator length) for proving the existence of surface subgroups in various classes of hyperbolic groups.
Groups are used by mathematicians to study symmetry. A group is just a collection of symmetries of an object. Examples include the geometric symmetries of a wallpaper pattern, or of a crystal structure, or of an Escher painting, or the algebraic symmetries associated to roots of polynomials. Mathematicians have studied groups intensively as abstract algebraic objects since the 19th century. In the 1980's M. Gromov proposed that we consider groups as geometric objects, and began to derive deep connections between the geometric and the algebraic properties of groups. One theme which emerged from Gromov's work is that the geometric properties which have deep algebraic consequences are not local properties, but rather coarse or "large scale". For example, and infinite ladder and an infinite straight line are not locally alike, but are large scale alike, and their symmetry groups will have many algebraic similarities. The PI investigates large scale versions of isoperimetric problems (area versus perimeter length problems) in groups, and also the geometry of coarsely negatively curved groups. These investigations help deepen our understanding of the nature of symmetry.
