Collapsing and relative hyperbolicity

摘要本课题涉及几何原点的相对双曲群，以及具有双边和下截界的塌缩。这些看似无关的主题自然出现在负截面曲率有限体积开放流形的结构和刚性研究中，其中（坍缩的）尖端对应于流形基本群中的外围子群。Belegradek最近证明负弯曲有限体积流形经常导致令人着迷的类相对双曲群，如紧实或复双曲流形中的超平面补的基本群。作为一个示例应用程序，Belegradek证明了这些组表现出mostow类型的刚性。这项工作的自然延续将是证明这些群是准等距刚性的。进一步，通过类比托莱多的结果，我们可以推测群不是剩余有限的，这就意味着非剩余有限双曲群的存在。Belegradek还计划通过Allcock-Carlson-Toledo最近的深入工作来研究更复杂的超平面补的相对双曲性，这些补作为代数曲面的模空间出现。负弯曲的尖端可以通过坍缩理论来研究，但只有当曲率为负时，才能得到全面的结构理论。本项目旨在研究在何种情况下可以构造具有良好几何特性的尖端截面，从而保证尖端截面的坍塌。这是建立在Belegradek与Kapovitch在消极挤压情况下的工作，以及Cheeger， Gromov， Fukaya和Rong的开创性工作的基础上的。其他项目涉及流形在低曲率界下的收敛性。为了改进Perelman的稳定性定理，计划研究允许DC（差分凹）结构的流形拓扑，并可能应用于光滑稳定性，以及与Alexandrov空间几何的自然联系。另一个项目是证明Yamaguchi和Fukaya的纤化定理的相对版本，并期望应用于非负弯曲开放流形中正常束到灵魂的坍缩和有限。坍缩理论研究在一定小尺度下呈现较低维的空间，其基本目标是在适当的曲率假设下了解坍缩区域的结构。双曲性是一种将几何洞察力带入刚性代数对象的方法。这两个概念结合在一起，因为双曲性可以防止在大部分空间上坍缩，这给了可以详细描述坍缩部分的希望。提出的研究可能会对几何和群论中的几个基本问题有所启发。

AbstractThis 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.