This project involves relatively hyperbolic groups of geometric origin, and collapsing with two-sided and lower sectional bounds. These seemingly unrelated themes naturally appear in studying structure and rigidity of finite volume open manifolds of negative sectional curvature, where the (collapsed) cusp ends correspond to peripheral subgroups in the fundamental group of the manifold. Belegradek recently showed that negatively curved finite volume manifolds often lead to fascinating classes of relatively hyperbolic groups, such as the fundamental groups of hyperplane complements in compact real or complex hyperbolic manifolds. As a sample application, Belegradek proved that these groups exhibit Mostow-type rigidity. A natural continuation of this work would be to show that the groups are quasi-isometrically rigid. Furthermore, by analogy with a result of Toledo, one could suspect that the groups are not residually finite, which would imply the existence of a non-residually-finite hyperbolic group. Belegradek also plans to investigate relative hyperbolicity for more intricate hyperplane complements that appear as moduli spaces of algebraic surfaces via recent deep works of Allcock-Carlson-Toledo. Negatively curved cusp ends can be studied via collapsing theory, yet a comprehensive structure theory is available only when the curvature is pinched negative. The project aims to study situations where one could construct cusp cross-sections with good geometric properties ensuring that cusp cross-sections collapse. This builds on Belegradek's work with Kapovitch in the negatively pinched case, as well as on seminal works by Cheeger, Gromov, Fukaya, and Rong. Other projects involve convergence of manifolds under a lower curvature bound. Aiming to improve on Perelman's stability theorem, it is planned to study topology of manifolds that admit DC (difference concave) structures, with possible applications to smooth stability, and natural connections to geometry of Alexandrov spaces. Another project is to prove relative versions of fibration theorems of Yamaguchi and Fukaya with expected applications to collapse and finiteness for normal bundles to souls in nonnegatively curved open manifolds.
Collapsing theory studies spaces that appear lower dimensional at a certain small scale, and a basic goal is to understand the structure of collapsed regions under suitable curvature assumptions. Hyperbolicity is a way to bring geometric insight into otherwise rigid algebraic objects. The two concepts fit together as hyperbolicity prevents collapsing on most of the space, which gives hope that the collapsed parts can be described in detail. The proposed research may shed light on several fundamental problems in geometry and group theory.
