The geometry, rigidity and combinatorics of spaces and groups with non-positive curvature feature

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Groups are fundamental abstract symbolic systems in mathematics arising from different branches of mathematics, physics, chemistry and computer science. For example, groups appear in the study of shapes of geometric objects, crystals and quasi-crystals, structure of roots of polynomials, cryptography, algorithm design etc. The study of finite groups, i.e. groups with finitely many elements, has reached a fairly mature stage, accumulating to a complete classification of finite simple groups. However, most infinite groups are fairly mysterious and hard to understand. In the 1980s, Gromov proposed a geometric approaches to group theory. One idea was to realize the mysterious group as a collection of symmetries of some geometric objects with interesting curvature properties, allowing us to study groups from the viewpoint of geometry. This has evolved into a very active field called geometric group theory. This proposal aims to study problems in the frontier of geometric group theory, and seeks applications to some long-standing problems in topology. The proposal also aims to provide resources for training graduate students and postdocs working in the area at the Ohio State University, with an emphasis on supporting under-represented early career stage mathematicians at OSU working in this filed.This proposal is concerned with rigidity and curvature properties of some infinite discrete groups from the viewpoint of geometric group theory, combined with ideas and techniques from ergodic theory, metric geometry, metric graph theory and combinatorial group theory. The project has two more specific research goals. The first is to make progress on a major conjecture on Artin groups, using a new strategy motivated from ideas in metric graph theory. The second is to understand fundamental forms of rigidity for discrete groups, namely quasi-isometric rigidity and measure equivalence rigidity. The project emphasizes the close connections between these forms of rigidity and the curvature properties of singular metric spaces and groups. Several classes of groups of fundamental importance are studied, including Artin groups, CAT(0) groups, graph products etc.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.