Relative hyperbolicity and nonpositive curvature

University Of Wisconsin-Milwaukee, Milwaukee WI

Investigators

Abstract

The scientific component of this proposal has two main parts. The first part is a program to understand relatively hyperbolic groups and CAT(0) spaces with isolated flats. Previous NSF funded work of the PI (including joint work with Bruce Kleiner) establishes a tight connection between these two subjects. The PI plans to continue a (single-authored) project that should substantially clarify and extend the foundations of relatively hyperbolic groups. The PI plans to continue a joint project with Kim Ruane to study the topology at infinity of a space with isolated flats.The PI also plans to continue a long term work in progress with Daniel Wise on the structure of groups acting on CAT(0) cube complexes. Their work on this project uses relatively hyperbolic groups in several ways and has already resulted in one NSF funded publication. They plan to cubulate many classes of relatively hyperbolic groups. In the second part of the research component, the PI plans to continue a program to study the lattice theory of the automorphism group of a CAT(0) simplicial complex. The PI has published NSF funded work on this subject (joint with Benson Farb), and plans to continue ongoing work with Christopher Connell. Additionally, the PI plans to publish an article, joint with Farb and Anne Thomas, surveying known results and open problems in this area. The PI is interested in the large scale geometry of spaces with nonpositive curvature. That is, they have a nontrivial mixture of negative curvature (like a saddle) and zero curvature (flat like a piece of paper), but no positive curvature (like the surface of a ball). More specifically, his research has focused on nonpositively curved spaces with isolated flats. These spaces are almost negatively curved in the sense that the flat, zero curvature, parts do not interact with each other. Thus techniques from the elegant theory of negatively curved spaces can sometimes be adapted to hold for spaces with isolated flats. In a separate project (joint with Benson Farb), the PI is studying the symmetries admitted by nonpositively curved spaces built by gluing together polygons along their edges. This project is an attempt to bridge the gap between the mathematical fields of geometric group theory and linear algebraic groups.