Research in Geometry and Topology

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。本研究计画主要研究自由群的对称性。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。该项目的目的是更好地了解这一现象。自由群Fn的外自同构群Out（Fn）的大规模几何的研究近年来取得了很大进展，但几个基本问题仍然是开放的。PI提出了一种策略来证明这个群的Farrell-Jones猜想。该策略是仿照相应的映射类组的结构的证明，考虑到额外的并发症，目前在出（Fn）。战略中的步骤提供了PI计划攻击的具体问题。更具体地说，目标是证明Out（Fn）具有有限的渐近维数，以更好地理解自由因子复合体的边界及其局部连通性，并通过证明Out（Fn）的自然电气化的距离公式来隔离Out（Fn）的大尺度几何不受双曲性支配的部分。在一个不同的方向，该项目试图构建无限多个准等距类型的双曲自由循环群与完全不可约单值，提供了与双曲三维流形的情况相反的现象，双曲三维流形在圆上纤维化，该奖项反映了NSF的法定使命，并被认为是值得通过使用基金会的智力价值和更广泛的影响力进行评估的支持审查标准。

The 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 research project 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, Lobachevski, Poincare and others, most recently to Gromov and Thurston. The goal of the project is to better understand this phenomenon. The award includes funds for graduate student support.The study of the large-scale geometry of the outer automorphism group Out(Fn) of the free group Fn has made great strides in recent years, but several fundamental questions are still open. The PI has proposed a strategy for proving the Farrell-Jones conjecture for this group. The strategy is modelled on the proofs of corresponding conjectures for mapping class groups, taking into account the extra complications present in Out(Fn). The steps in the strategy provide concrete questions the PI plans to attack. More specifically, a goal is to prove that Out(Fn) has finite asymptotic dimension, to better understand the boundary of the complex of free factors and its local connectivity, and to isolate the part of large scale geometry of Out(Fn) not governed by hyperbolicity by proving a distance formula for a natural electrification of Out(Fn). In a different direction, the project attempts to construct infinitely many quasi-isometry types of hyperbolic free by cyclic groups with fully irreducible monodromies, providing a phenomenon opposite from the situation for hyperbolic 3-manifolds that fiber over the circle with pseudo-Anosov monodromies.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.