Research in Geometry and Topology

[摘要]获奖：DMS 1607236，首席研究员：Mladen bestvina代数与几何的相互作用是数学中的经典主题之一。传统上，人们通过几何物体的对称性来研究它们。在几何群论中，情况是相反的：一个人从一个代数对象（例如，另一个组）的一组对称性开始，并构造一个具有相同对称性的几何对象。这一建议的重点是研究一个自由群的对称性。令人惊讶的是，它的几何在很大程度上是由双曲几何控制的，这种几何可以追溯到高斯、洛巴切夫斯基、庞加莱和其他人，最近的是格罗莫夫和瑟斯顿。该项目的目标是更好地理解这一现象。近年来，对n个生成子上自由群的外自同构群的大尺度几何的研究取得了很大进展，但仍有几个基本问题有待解决。首席研究员提出了一种策略来证明诺维科夫和法雷尔-琼斯的猜想。该策略基于映射类群的相应猜想的证明，并考虑到Out（F_n）中存在的额外复杂性。该战略中的步骤提供了PI计划解决的具体问题。更具体地说，目标是证明Out（F_n）具有有限渐近维数，更好地理解自由分裂复合体的边界，并研究从Out（F_n）映射到投影复合体的合适乘积的轨道在多大程度上不是准等距嵌入。其他问题包括映射类群中的凸紧性，标准Out(F_n)-复合体的一致双曲性，以及裤子复合体的立方化。

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.