This proposal has two main scientific components. The first part is a project to understand relatively hyperbolic groups and CAT(0) spaces with isolated flats, two topics that are tightly connected by previous work of the PI with Bruce Kleiner. The PI plans to continue a joint project with Kim Ruane to study the topology of the boundary of a space with isolated flats and its relation to group splittings. The PI also plans to continue work in progress with Daniel Wise on a-T-menability and its relation to CAT(0) cubical complexes. In particular, they plan to cubulate many classes of hyperbolic and relatively hyperbolic groups. In second main component, the PI plans to continue a program to study the lattice theory of the automorphism group of a simplicial complex, in a joint project with Benson Farb. This program is at the frontier between geometric group theory and the theory of discrete subgroups of Lie groups and linear algebraic groups, and generalizes recent work of Bass--Lubotzky on tree lattices.
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.
