Hyperbolic manifolds and groups

A 3-manifold is a space where an object can move around in three distinct perpendicular directions. The universe is a three-manifold whose global structure is not yet understand. Thanks to transformative work around the turn of the century, it is known 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. The project includes problems pursued in collaboration with current and recent graduate students mentored by the PI. The project also supports the PI’s leadership efforts in building stronger mentoring in his department, nurturing the mathematical community in the Philadelphia area, and training junior mathematicians through a national graduate conference. Mathematically, this project seeks to make progress on several important open questions about hyperbolic 3-manifolds and their fundamental groups, with emphasis on effective computation and large-scale structure. One question involves quantitative control on the change in geometry under Dehn surgery, including applications to the cosmetic surgery conjecture that are coded into software for the mathematical community. The second question involves identifying the Margulis constant and understanding the structure of Kleinian groups generated by two short elements. The third question involves relationships between the rank and genus of 3-manifolds. The fourth question involves a coarse understanding of the fixed-point properties of pseudo-Anosov maps on surfaces, with applications to invariants of knots and 3-manifolds. The fifth question involves a quantitative understanding of special covers of 3-manifolds. 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.