Aspects of non-positive curvature

Ohio State University, The, Columbus OH

Investigators

Abstract

Abstract Award: DMS 1510640, Principal Investigator: Jean-Francois R. Lafont The main goal of this project is to study spaces of negative (or more generally, non-positive) curvature. Negatively curved spaces are ones which, at every point and every pair of directions, look like the surface of a saddle. While this type of space might sound odd, it is in fact pervasive in mathematics. For these spaces, there is a tight link between the geometry (quantitative features of the space) and the topology (qualitative features of the space). In addition, these spaces have some very interesting dynamical systems associated to them, where one can study how objects move over time. As a result, the study of these spaces provides a melting pot for different viewpoints and methods, making for a rich field of study. This project plans on using these three different approaches (geometry, topology, dynamics) to shed some light on these spaces. The Principal Investigator (PI) plans to work on a series of projects that are centered around the notion of non-positive curvature. The PI's projects loosely fall into three broad categories: geometry, topology, and dynamics. On the geometric side of things, the PI plans on focusing on the distinction, for manifolds, between Riemannian non-positive curvature and metric non-positive curvature. The PI also proposes to study discrete versions of Gromov's systolic inequality. Another project involves studying the relation of almost-isometry for groups and spaces. On the topological side of things, the PI plans on giving combinatorial formulas for the topological K-theory of 3-dimensional hyperbolic reflection groups. The PI will also classify Gromov hyperbolic groups with Sierpinski curve boundary. Another project concerns hyperbolization and the sign of the Euler characteristic. Finally, the PI plans on studying the topology of Anosov splittings. For the dynamical aspects of this proposal, the PI wants to establish marked length rigidity for Fuchsian buildings. He also wants to show that certain aspherical manifolds cannot support Anosov diffeomorphisms. Another project is to show the geodesic flow on locally CAT(-1) spaces has the specification property.