Mesoscopic Topology in Geometric group theory

University Of Illinois At Chicago, Chicago IL

Investigators

Abstract

A central theme in mathematics for centuries has been the interaction between algebra and geometry. The usual direction is to study the set of symmetries of a geometric object of interest. In geometric group theory, this becomes a two-way street in that algebraic objects (such as groups) are considered as geometric objects in their own right. Hyperbolic geometry is a subject going back to work of Bolyai, Gauss and others in the 19th Century, but it also plays a central role in modern geometry, particularly due to the influence of Thurston and Gromov. This project centers around a central question in geometric group theory, the Cannon Conjecture, about the difference (in three dimensions) between classical hyperbolic geometry and the coarse notion due to Gromov, in the presence of a large group of symmetries. The principal investigator (PI) will expand the work of the MSCS Undergraduate Research Laboratory at the University of Illinois at Chicago, extending the reach of research projects to a wide audience of students there. In recent work of the PI with Haissinsky, Manning, Osajda, Sisto and Walsh, the PI has developed a theory of “drilling” residually finite hyperbolic groups with two-sphere boundary. This reduced the Cannon Conjecture to a relatively hyperbolic version, which should be more tractable since it corresponds to manifolds with boundary. In this project, the PI will extend this work in three directions. In work with Walsh, the PI will use tools from drilling to develop a coarse version of Calegari-Gabai’s Shrinkwrapping, and prove that in a residually finite hyperbolic group with two-sphere boundary, a surface subgroup is either quasi-convex, or a virtual fiber. In work with Manning, the PI will extend the work from the Drilling project in order to drill a graph instead of a curve, thereby reducing the Cannon Conjecture to a conjecture of Kapovich-Kleiner about hyperbolic groups with Sierpinski Carpet boundary. With Wilton, the PI will develop a notion of coarse sectional curvature, with applications to coherence and local quasi-convexity of certain hyperbolic groups, particularly those with Sierpinski Carpet boundary. 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.