Hyperbolic Manifolds and Their Moduli Spaces

University Of Wisconsin-Madison, Madison WI

Investigators

Abstract

In mathematics one is often faced with the problem of understanding a given geometric object. Surprisingly often, it is advantageous to deform the geometry of this object in order to better understand it. That is, one may understand something geometric by understanding how one may change it. The prototypical example of this is Riemann's moduli space of curves, which is a space each of whose points represents a two-dimensional object called a Riemann surface. This space captures all of the ways one may deform these surfaces. The proposed research will explore the geometry of this space, its analogs in spaces of deformations of certain three-dimensional spaces called hyperbolic manifolds, and their interrelationships. It also contains suitable sub-projects for graduate students. The project falls into two parts: the deformation theory of hyperbolic three-manifolds; and the study of geometric and algebraic properties of subgroups of mapping class groups of surfaces, their associated surface bundles, and profinite completions. The first is principally concerned with understanding a certain function called the skinning map, which was discovered and studied by Thurston in his work on the geometrization of three-manifolds, and which measures the effects of deformations of hyperbolic three-manifolds. A better understanding of this function sheds light on the geometry of three-manifolds and its relation to topology. In particular, the PI will continue an ongoing project with collaborators Bromberg and Minsky to establish effective bounds on the diameter of this map and its relation to the notion of renormalized volume. The second part is concerned with the study of the geometry of subgroups of mapping class groups and their relation to the geometry of four-manifolds in collaboration with Leininger. The second part will also continue the PI's work in profinite aspects of mapping class groups, with a view toward Ivanov's congruence subgroup problem. 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.