Research in Geometry and Topology

代数和几何之间的相互作用是数学的经典主题之一。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。本研究项目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受到双曲几何的支配，这种几何可以追溯到高斯、洛巴切夫斯基、庞加莱和其他人，最近的一次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。该奖项包括研究生资助资金。近年来，自由群FN的外自同构群OUT(FN)的大规模几何研究取得了很大进展，但仍有几个基本问题有待解决。PI提出了一种策略来证明这个群体的Farrell-Jones猜想。该策略以映射类群的相应猜想的证明为模型，考虑了OUT(FN)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(Fn)有有限的渐近维度，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)的自然带电的距离公式来分离Out的大尺度几何(Fn)中不受双曲性支配的部分。在不同的方向上，该项目试图通过具有完全不可约的单边线的循环群来构造无限多个准等距自由的双曲线型，提供了与具有伪Anosov单边线的环上的双曲线3-流形的情况相反的现象。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势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.