It is often useful to understand a class or sequence of spaces through its geometric limits. The PI proposes five projects applying such asymptotic methods to topology, geometry and group theory. First, the PI will study the large-scale geometry of closed hyperbolic 3-manifolds whose fundamental groups can be generated by a fixed number of elements. Second, the PI will study algebraic and geometric limits of convex cocompact handlebodies and their relation to the extension of homeomorphisms of a 3-manifold's boundary into its interior. In group theory, the PI hopes to investigate the quasi-isometric rigidity of mapping tori of certain free group automorphisms and the asymptotic growth of the index of the intersection of all index n subgroups of a group. Finally, the PI hopes to promote and expand the exciting new topic of `invariant random subgroups' of Lie groups, which he and Abert et al used to control the growth of Betti numbers of higher rank locally symmetric spaces.
The study of geometric shapes of 2 and 3 dimensions has rapidly evolved in the past 40 years, partly due to revolutionary work of William Thurston in the 1970's. Thurston was one of the first to realize the particular utility of 'geometric degenerations' in these low dimensions. A geometric degeneration can be thought of as a deformation of a shape that possibly changes its type at the end: for instance, a sphere with a shrinking radius degenerates to a point and a vertical cylinder with fixed height and shrinking circumference degenerates to a line segment. The goal of the proposed project is to use geometric degenerations to analyze a variety of low-dimensional shapes, especially 3-dimensional manifolds, of which our universe is an example.
