Hyperbolic Geometry and the Mapping Class Group

University Of Utah, Salt Lake City UT

Investigators

Abstract

Two and three-dimensional spaces are some of the most fundamental objects studied by mathematicians. One approach to study them is to realize them as geometric objects and then use the geometry to study the spaces. The three most common geometries are first, the classical Euclidean or flat geometry, and then the non-Euclidean: spherical and hyperbolic geometries. It has been known since the late 1800s that most two-dimensional spaces, or in other words: surfaces, exhibit hyperbolic geometry. Thurston's groundbreaking work in the 1970s showed that "most" three dimensional spaces, called three-manifolds, also have hyperbolic geometry. This is the motivation for this project's focus on spaces with hyperbolic geometry. It includes many sub-projects suitable for training graduate students to work in this area. This project will study several questions on the geometry of hyperbolic three-manifolds building on PI's prior work. One of the central topics is the study of "renormalized volume". The PI and his collaborators recently initiated a study of the gradient flow of the renormalized volume function which has shown to be an extremely useful tool for understanding the geometry of hyperbolic three-manifolds. In another direction, the PI and his collaborators will continue their work in geometric group theory to understand actions of the mapping class group on CAT(0) cube complexes. 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.