Combinatorics, Models, and Bounds in Hyperbolic Geometry

Brown University, Providence RI

Investigators

Abstract

The notion of "bounded combinatorics" in the complex of curves on a surface controls geometry in a variety of settings. Understanding and bounding how simple closed curves on a surface can project to subsurfaces provides tools to construct models for hyperbolic 3-manifolds, classify Weil-Petersson geodesics, and elucidate the fine topological structure in boundaries of deformation spaces. Indeed, the existence and uniqueness of hyperbolic structures on 3-manifolds gives little information about their geometric features and their connection to topological properties of the manifold. Through his expansion of the use of the bi-Lipschitz Model Theorem (of the PI with Canary and Minsky), the PI will explore geometric models for arbitrary closed hyperbolic 3-manifolds and connect their structure to combinatorial features of the manifold. This ongoing project with collaborators Minsky, Namazi and Souto will produce explicit models for Heegaard splittings, and a new weak geometrization for 3-manifolds arising as infinite gluings with bounded combinatorics. Further, the PI will apply coarse methods in the study of the mapping class group via the curve complex and its associated "hierarchy paths" to understand the large scale structure of the Weil-Petersson metric on Teichmueller space, a fundamental object whose large scale geometry remains mysterious despite many investigations. Finally, the PI will exhibit further features of the deformation space of a hyperbolic 3-manifold, generalizing our study of the local topology of deformation spaces with Bromberg, Canary, and Minsky, and undertaking generalized studies of central compactness theorems for deformation spaces in the context of the curve complex. In mathematics, the study of dynamical systems seeks to describe chaotic phenomena in simple terms. Sometimes dynamical systems can exhibit a kind of rigidity, where a small tweak or perturbation does not affect the long term behavior of the system. In the context of understanding our own three-dimesnsional universe and what kind of structures three-dimensional spaces can have, structures give rise to these rigid dynamical systems. The recent work of Grisha Perelman solving the famous Poincare Conjecture has ensured this study applies to virtually all three-dimensional spaces. When a space is rigid, one can understand it completely via "coarse" information, via so-called "models." In a recent result of the PI with R. Canary and Y. Minsky, such models were used to classify all constantly negatively curved, or "hyperbolic" three-dimensional spaces of infinite volume that are tame in a certain sense. The classification result solved a long-standing conjecture of William Thurston, and opened the door to developing a more detailed and complete picture of geometries on manifolds previously considered understood. The groundwork is in place for a fundamental investigation of algebraic and topological properties of all spaces of 3-dimensions and how these properties interrelate.