The Geometry of Hyperbolic 3-Manifolds

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Surfaces are fundamental objects in geometry and typically come in three flavors: flat, round, or saddle shaped, geometries more technically referred to as Euclidean, spherical, or hyperbolic. Three-dimensional objects, too, typically come in one of a few flavors, including Euclidean, spherical, and hyperbolic, plus a few others. In both two and three dimensions, the hyperbolic geometries are the most common and currently the most studied. This project is concerned with the geometry of 3-dimensional hyperbolic spaces. Its broad aim is to build complete models of the geometry from fundamental building blocks, completing a long line of inquiry in the subject. In addition to its core research objectives, the project serves the profession and the advancement of science through plans for continued mentoring of students, advocacy for underrepresented groups, and several outreach and service activities. The project is concerned with the geometry of hyperbolic 3-manifolds and their deformations. The project is centered on a study of Thurston's skinning map, which is a holomorphic function on the Teichmüller space associated to a hyperbolic 3-manifold of infinite volume. The size and shape of the image of this map has implications for the underlying geometry of the hyperbolic manifold, and the project aims to bound the size of the image in terms of the topology of the underlying manifold's boundary, without any dependence on the topology of the underlying manifold. Using this, the project aims to construct uniform models of hyperbolic 3-manifolds. The techniques involved in the approach are of wider interest, and the project aims to use these techniques to establish new theorems on the geometric inflexibility of hyperbolic manifolds. The project also aims to establish new universal hyperbolic Dehn filling theorems. The project's new techniques will also be aimed at establishing the existence of hyperbolic integral homology spheres of large injectivity radius.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.