Pallavi Dani is the recipients of an NSF grant on “Coarse Geometry and Rigidity in Coxeter and Hyperbolic Groups”.
The new research project aims to explore the intricate structures of groups, which are fundamental not only in mathematics but also in various scientific fields like physics, chemistry, computer science, and cryptography. These groups can manifest as collections of functions, within dynamical systems, or as symmetries of objects, including physical entities like crystals. The project intends to employ tools from geometry and topology to better understand the classification and subgroup structures of two significant classes of groups. Alongside the core research, the project will provide mathematical training across multiple levels—undergraduate, graduate, and postdoctoral—and engage in outreach activities targeting local middle and high school students as well as regional undergraduates.
The research focuses on Coxeter groups and hyperbolic groups through the lens of geometric group theory, addressing central challenges such as quasi-isometric classification and the rigidity of finitely generated groups. While Coxeter groups have been studied extensively from various angles, their rigidity properties remain relatively underexplored. This project seeks to fill that gap by developing new quasi-isometry invariants, like the hypergraph index, and examining these in diverse subclasses, including PM type Coxeter groups and thick Coxeter groups. Additionally, the project will investigate the lesser-known subgroup structures of hyperbolic groups, particularly focusing on distortion, Dehn functions, and boundary maps, with the goal of constructing new examples that could help test current mathematical conjectures.