Collapsing and relative hyperbolicity

塌陷和相对双曲性

• 批准号: 0804038
• 负责人: Igor Belegradek
• 金额: $ 15.18万
• 依托单位: Georgia Tech Research Corporation
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2008
• 资助国家: 美国
• 起止时间: 2008-09-01 至 2012-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0804038&HistoricalAwards=false
• 关键词: Collapsing; relative; hyperbolicity

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.

本项目涉及几何起源的相对双曲群，以及具有双边和下截面界的坍缩，这些看似无关的主题自然地出现在研究具有负截面曲率的有限体积开流形的结构和刚性时，其中（崩溃）尖端对应于流形的基本群中的外围子群。Belegradek最近表明，负弯曲的有限体积流形往往导致到迷人的类相对双曲群，如基本群的超平面补在紧凑的真实的或复杂的双曲流形。作为一个例子的应用，Belegradek证明，这些群体表现出Mostow型刚性。这项工作的一个自然的延续是证明这些群是拟等距刚性的。此外，通过与托莱多的结果类比，人们可以怀疑这些群不是剩余有限的，这意味着存在一个非剩余有限的双曲群。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，这些超平面补作为代数曲面的模空间出现。负弯曲的尖点末端可以通过塌缩理论来研究，但是只有当曲率为负时，才有全面的结构理论。该项目旨在研究可以构建具有良好几何特性的尖点横截面的情况，以确保尖点横截面崩溃。 这建立在Belegradek的工作与Kapovitch在负捏的情况下，以及开创性的工作由Cheeger，格罗莫夫，福谷，荣。其他项目涉及收敛的流形下的曲率下限。为了改进佩雷尔曼的稳定性定理，计划研究允许DC（差分凹）结构的流形的拓扑，可能应用于光滑稳定性，以及与亚历山德罗夫空间几何的自然联系。另一个项目是证明山口和福谷的纤维化定理的相对版本，预期应用于非负弯曲开流形中灵魂的法丛的坍塌和有限性。坍塌理论研究在一定小尺度下出现的低维空间，其基本目标是理解合适曲率假设下坍塌区域的结构。双曲性是一种将几何洞察力带入其他刚性代数对象的方法。这两个概念可以结合在一起，因为双曲面可以防止大部分空间的塌陷，这给了我们希望，可以详细描述塌陷的部分。所提出的研究可能揭示几何和群论中的几个基本问题。