The PI is interested in the large scale geometry of groups and complexes. One project involves analyzing the asymptotic behavior of filling invariants of groups. In collaboration with Martin Bridson, Max Forester and Krishnan Shankar, the PI is investigating the first and second order Dehn functions of groups. In joint work with Max Forester and Krishnan Shankar, the PI is exploring what types of functions can arise as first or second order Dehn functions of subgroups of CAT(0) groups, and of hyperbolic groups. Another project of the PI and John Crisp, Anton Kaul and Jon McCammond involves generalized Garside structures and associated complexes for Artin groups. Another project, comprised of separate collaborations of the PI with John Crisp and with Jon McCammond, focuses on non-positive curvature and dimension in group theory, and on the connections between large scale notions and local notions of non-positive curvature. A number of other projects include investigations into conjugacy and isomorphism of generalized Baumslag-Solitar groups, ends of amalgams, distortion of subgroups of hyperbolic groups, and the existence of surface subgroups 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.
