Rigidity, Volume, and Combinatorics in Hyperbolic Geometry

Brown University, Providence RI

Investigators

Abstract

For much of the twentieth century, the challenge to describe the shape of three dimensional spaces was viewed as an algebraic problem: indeed it is through algebra that we first distinguish the (topological) structure of a sphere from that of a doughnut. Work of William Thurston brought the geometry of such spaces more clearly into view as a central feature to explore. The triumph of this approach was Perelman's proof of Thurston's geometrization conjecture, that each three-dimensional manifold could be naturally broken up into pieces, each with a uniform geometry. The study of these geometries is frequently reduced, via a notion of rigidity, to considering simple combinatorial structures arising from loops on surfaces. The proposed research will explore how these structures predict volume, diameter, length and other aspects of the geometry, relating to notions from quantum physics. A longstanding connection between volume of hyperbolic three-manifolds and Weil-Petersson distance relied on a combinatorial comparison via the pants graph which organizes maximal multicurves on a surface, yet other connections were known to arise from work of Witten via renormalized volume. Recent work of Schlenker made this connection explicit in the context of quasi-Fuchsian manifolds, and PI's proposed work will develop this idea further to explicitly relate fibered 3-manifolds and translation distiance. More generally, a primary project in the proposed research is to solidify and extend the connection between combinatorics and geometry in closed manifolds, and to continue to investigate the structure of deformation spaces of Kleinian groups with these tools. As an example of the power of these techniques, bi-Lipschitz models for 'random' Heegaard splittings provide a full solution (with Rivin and Souto) to the conjecture of Dunfield and Thurston that random Heegaard splittings are almost surely hyperbolic and their volume grows linearly. Finally, the PI will pursue projects with Minsky, and with Modami, Leininger and Rafi on the role of combinatorics in the geometry of geodesics in the Weil-Petersson metric, investigating disparities between Teichmuller and Weil-Petersson geodesics in terms of unique ergodicity.