Geometry and topology of hyperbolic 3-manifolds

University Of Illinois At Chicago, Chicago IL

Investigators

Abstract

The project explores the geometry and topology of three-dimensional hyperbolic manifolds, those which admit a Riemannian metric with negative sectional curvatures. Some of the most fundamental questions about hyperbolic 3-manifolds concern the extent to which their properties are inherited in, or, conversely, may develop among covering spaces of finite degree. The famous virtual Haken and virtual fiberering conjectures answer this description, for example, as do the questions below. Given a hyperbolic 3-manifold with a special topological property, for instance a knot complement, with how many others that have this same property does it share a finite-degree cover? Or for a fixed hyperbolic 3-manifold M and a family of finite-degree covers {M_n}, does the rank of the fundamental group of M_n grow linearly with the covering degree? This relates to the fixed price question in the study of topological dynamics. The PI intends to continue his attacks on these and other questions, such as the possibility of embedding in right-angled Artin groups. He will also further describe the topology of hyperbolic manifolds with low volume. The objects of study here, 3-manifolds, are spaces which look, in a neighborhood of each point, like the familiar three-dimensional space in which we live. Although their definition allows quite a bit of flexibility, 3-manifolds share with 2-manifolds (surfaces) the property that each admits a unique best metric, a way of measuring the distance between two points in the space. This property of low-dimensional manifolds, known in dimension 3 as Thurston's geometrization conjecture and proven recently by V. Perelman, has motivated an enormous body of work that uses geometry as a tool for understanding 3-manifolds, and the project follows this broad theme. The study of 3-manifolds has benefited from its interaction with many different fields of mathematics, and the project additionally draws from questions and techniques in the study of topological dynamics, geometric flows, geometric group theory, and others. The project will help us better understand the classification of 3-manifolds and the relationships between them. This in turn has applications in such diverse fields as cosmology (in determining the shape of the universe, for example), biology (in understanding the knotting of DNA), and computer science (through connections with families of graphs that certain families of 3-manifolds coarsely resemble), among others.