Research in Geometry and Topology

摘要奖：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，情况是相反的：一个人从一个代数对象(例如，另一个群)的一组对称开始，然后构造一个具有相同对称性的几何对象。这一提议侧重于研究自由群的对称性。令人惊讶的是，它的几何在很大程度上受到双曲几何的支配，这可以追溯到高斯、洛巴耶夫斯基、庞加莱和其他人，最近的一次是格罗莫夫和瑟斯顿。这个项目的目的是为了更好地理解这一现象。关于n个生成元上自由群的外自同构群的大规模几何的研究，通常表示为(F_N)，近年来取得了很大进展，但仍有几个基本问题有待解决。首席研究员已经提出了一种策略来证明这一组的Novikov和Farrell-Jones猜想。该策略基于映射类群的相应猜想的证明，并考虑了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，目的是证明Out(F_N)有有限的渐近维度，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多大程度上不是准等距嵌入。另外还讨论了映射类群的凸余紧性的刻画，标准输出(F_N)-复形的一致双曲性，以及裤子复形的立方问题。

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.