Research in Geometry and Topology

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。本课题主要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是由双曲几何控制的，这种几何可以追溯到高斯、罗巴切夫斯基、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。该奖项包括研究生支持基金。自由群Fn的外自同构群Out（Fn）的大尺度几何研究近年来取得了很大进展，但仍有几个基本问题有待解决。PI提出了一种策略来为这个群体证明法雷尔-琼斯猜想。该策略以映射类群的相应猜想的证明为模型，并考虑到Out（Fn）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，我们的目标是证明Out（Fn）具有有限的渐近维数，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）自然电化的距离公式来隔离Out（Fn）不受双曲控制的大尺度几何部分。在另一个方向上，该项目试图用完全不可约单环群构造无限多准等距离型的双曲自由，提供了一种与双曲3-流形的情况相反的现象，双曲3-流形在圆上纤维具有伪anosov单环。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.