Infinite Discrete Groups

Cornell University, Ithaca NY

Investigators

Abstract

This project has two strands: algebraic versus geometric invariants of groups, and exploring non-positively curved groups. Both fall within the orbit of Gromov's study of infinite discrete groups through large-scale geometry, heavily driven by parallels with Riemannian geometry. The first focuses on filling invariants, which concern the geometric features of discs spanning loops ("soap film geometry") in spaces associated with groups. The second concerns how notions of non-positive curvature in group theory interrelate and the wildness present in non?positively curved groups, particularly in their conjugacy problems and subgroups. The work falls within the subject of Geometric Group Theory, which concerns infinite groups and the spaces on which they act. Its roots lie in the early 20th century, particularly in low-dimensional and algebraic topology. It has taken off since the late 1980s when it became clear that geometric features of the spaces involved have profound repercussions for the algebraic structure of the groups. Rich interactions have ensued with, for example, Riemannian geometry, probability, Lie groups, dynamical systems, ergodic theory, combinatorics, computer science and logic. In addition to the research work, a course in Geometry and Topology will be deigned for undergraduates whose studies who are not primarily concentrated in mathematics.