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.
The most important outcome of this project is the creation of the centered dual decomposition, an organizing principle for faces of a two-dimensional Delaunay tessellation that have a certain pathology. The context here is the "meshing problem": given a (say) finite, plane set S, find a polygonal decomposition of its convex hull with the original set S as its vertex set. Meshing is a technique used across mathematics and its applications for analyzing continuous phenomena using sample points. The Delaunay tessellation is a classical (proposed in 1934 by Boris Delaunay) but still much-used solution to the meshing problem. It is determined by the empty circumcircles condition: the vertex set of every face must lie on a planar circle that bounds a disk containing no other points of S. The Euclidean Delaunay tessellation is dual to the Voronoi tessellation of S which has a face for every element s of S, consisting of points closer to s than any other point of S. Motivated by an application to volumes of hyperbolic 3-manifolds, we consider Delaunay tessellations of sets in the hyperbolic plane, the two-dimensional geometry where the uniqueness aspect of Euclid's parallel postulate fails. One outcome of the project is a construction of the Delaunay tessellation of an infinite subset of hyperbolic space that is invariant under a collection of hyperbolic isometries; i.e. rigid motions. This is the relevant setting for our construction. Our goal is to produce a lower bound on the area of Delaunay faces in terms of a bound on their edge lengths. This is facilitated by the fact that since each face is cyclic, i.e. inscribed in a circle, its area is determined by its side length collection. But it turns out that the area of cyclic polygons is "increasing in edge lengths" only among those that are centered. We say cyclic polygon is centered if its interior contains the center of its circumcircle. So it is the non-centered cells that are pathological for our purposes. The issue we face is a version of the "skinny triangles" problem encountered in numerical analysis and computational geometry. In that setting Ruppert's algorithm yields an effective method for mitigating the skinny triangles problem by adding points to the given set S, but in ours it is important that S remain unchanged. The novelty of our approach is that we do not attempt to "fix" individual cells. Instead, we observe that non-centered Delaunay faces have dual Voronoi vertices contained in edges that are non-centered in a different sense. Components of the union of such Voronoi edges are trees (in the mathematical sense), and grouping Delaunay faces dual to the vertices of such trees yields the centered dual cells that we use to realize our goal. The desired application to volumes of hyperbolic 3-manifolds remains in progress, but we found initial success in another direction, describing the maximum of injectivity radius among all points of all finite-area hyperbolic surfaces of a fixed topological type. (This generalized earlier work of C. Bavard.) We hope the centered dual will find broader application to other versions of the meshing problem featuring a preferred vertex set. An outcome of the "commensurability" portion of the project that we hope will find further use is a new technique for constructing hidden symmetries of manifolds: symmetries of finite-degree that do not descend to the manifolds themselves. Interest in hidden symmetries goes back at least to work of A. Borel and was further spurred by G. Margulis' characterization of arithmetic manifolds in terms of them. In joint work with Eric Chesebro, we show in a family of examples that we previously constructed how to construct hidden symmetries of a manifold that decomposes into pieces using special hidden symmetries (called hidden extensions) of the pieces. An outcome of the "rank" portion of the project that we find worth highlighting is actually a question. It arises from joint work with Stefan Friedl and Stefano Vidussi, where we proved that certain infinite families of 3-manifolds ("cyclic covers" associated to an infinite-order first cohomology class of a fixed manifold) have positive rank gradient if and only if they have positive Heegaard gradient. Rank of fundamental group and Heegaard genus are invariants that each measure a sort of complexity of a 3-manifold. Their gradients measure how these invariants grow among members of a family of 3-manifolds. W. Haken asked if every 3-manifold has Heegaard genus equal to the rank of its fundamental group? The answer is no -- most recently T. Li gave hyperbolic examples -- but one may ask a weaker question: among any family of 3-manifolds, must the rank and Heegaard gradients vanish simultaneously if at all? There is evidence in this context of coherent behavior, including the result of the PI, Friedl and Vidussi mentioned above.
