Hyperbolic Manifolds and Their Groups

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A three-manifold is a space where an object can move around in three distinct perpendicular directions. The universe that we inhabit is a three-manifold whose global structure we do not yet understand. Thanks to powerful theorems by Thurston, Perelman, and Mostow, we do know that the geometry of a manifold (measurements of angles, distances, and curvature) is closely tied to its large-scale structure. What is missing at this point is a quantitative understanding of how geometry and large-scale topology determine one another. This project seeks quantitative information of this nature. It contains suitable sub-projects for graduate students. More specifically, this project seeks to make progress on several fundamental questions involving the geometry of negatively curved three-manifolds and their fundamental groups. One question involves quantitative control on the change in geometry under Dehn surgery, including applications to the cosmetic surgery conjecture. A second question involves understanding Kleinian groups in which two independent elements move a point by a small distance, with applications to bounding the Margulis constant. A third question involves a quantitative understanding of the way in which three-manifold groups act on CAT(0) cube complexes, with an eye toward developing tools for the Cannon conjecture.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.