Research in Geometry and Topology

几何与拓扑研究

• 批准号: 1607236
• 负责人: Mladen Bestvina
• 金额: $ 34.5万
• 依托单位: University of Utah
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2016
• 资助国家: 美国
• 起止时间: 2016-09-01 至 2020-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1607236&HistoricalAwards=false
• 关键词: Research; Geometry; Topology

AbstractAward: DMS 1607236, Principal Investigator: Mladen BestvinaThe interplay between algebra and geometry is one of the classical themes in mathematics. Traditionally, one studies geometric objects via their symmetries. In geometric group theory the situation is reversed: one starts with a group of symmetries of an algebraic object (for example, another group) and constructs a geometric object with the same symmetries. This proposal focuses on the study of symmetries of a free group. Surprisingly, its geometry is to a large degree governed by hyperbolic geometry that goes back to Gauss, Lobacevski, Poincare and others, most recently to Gromov and Thurston. The goal of the project is to better understand this phenomenon.The study of the large-scale geometry of the outer automorphism group of a free group on n generators, usually denoted Out(F_n) has made great strides in recent years, but several fundamental questions are still open. The principal investigator has proposed a strategy for proving the Novikov and Farrell-Jones conjectures for this group. The strategy is modeled on the proofs of corresponding conjectures for mapping class groups, taking into account the extra complications present in Out(F_n). The steps in the strategy provide concrete questions the PI plans to attack. More specifically, a goal is to prove that Out(F_n) has finite asymptotic dimension, to better understand the boundary of the complex of free splittings, and to study the extent to which the orbit map from Out(F_n) to a suitable product of projection complexes fails to be a quasi-isometric embedding. Additional questions include characterization of convex cocompactness in mapping class groups, uniform hyperbolicity of standard Out(F_n)-complexes, and cubing the pants complex.

AbstractAward：DMS 1607236，首席研究员：Mladen Bestvina代数和几何之间的相互作用是数学中的经典主题之一。传统上，人们通过几何对象的对称性来研究它们。在几何群论中，情况正好相反：人们从一个代数对象的一组对称性（例如，另一组）开始，并构造一个具有相同对称性的几何对象。这个建议的重点是研究一个自由群的对称性。令人惊讶的是，它的几何形状在很大程度上受双曲几何的支配，这可以追溯到高斯、罗巴切夫斯基、庞加莱等人，最近是格罗莫夫和瑟斯顿。近年来，关于n个生成元上的自由群的外自同构群（通常记为Out（F_n））的大规模几何的研究取得了很大的进展，但仍有一些基本问题没有解决。首席研究员提出了一个策略，以证明诺维科夫和法雷尔-琼斯定理为该组。该策略是建立在映射类组的相应结构的证明上的，考虑到Out（F_n）中存在的额外复杂性。战略中的步骤提供了PI计划攻击的具体问题。更具体地说，我们的目标是证明Out（F_n）具有有限的渐近维数，更好地理解自由分裂复形的边界，并研究从Out（F_n）到投影复形的适当乘积的轨道映射在何种程度上不是拟等距嵌入。其他问题包括映射类群中凸余紧性的刻画，标准Out（F_n）-复形的一致双曲性，以及裤子复形的立方化。