Intrinsic rigid structure in groups and surfaces

奖项：DMS 2005368，主要研究者：Spencer D. Dowdall这个项目涉及数学的两个基础分支：群论和低维拓扑/几何。群是一个代数框架，它编码对象的对称性，例如空间的刚性运动，魔方的配置或分子的对称性。 低维拓扑学和几何学关注的是空间本身的结构，比如地球表面或我们生活的宇宙，以及固有的几何特征，比如曲率、距离和体积。其中一种方法是通过参数空间的概念（如给定对象上支持的几何结构，或该对象中点的配置）及其对称群来联系这些主题。这个项目将集中在两个重要的类的例子在这些上下文中：群扩展，这是建立新的群体的旧的方式，和映射类组的表面，这是对称群的参数空间的双曲结构的表面。该项目的目的是确定和研究关键的结构特征，特别是在扰动或重新参数化下不变的刚性特征，以及随机构造中出现的一般特征。本计画为研究生提供研究训练的机会，具体而言，本计画将探讨自由群与曲面自同构的几何与动力学方面，并探讨这些自同构如何与相关的群扩张与几何结构的模空间互动。首先，本项目将研究曲面群和自由群的循环扩张。这就需要使用转向三角研究低膨胀伪Anosov表面同胚，并发展一个平行的理论转向三角自由循环群，将照亮功能的自由群自同构。该项目还将为自由循环群引入一种新的通用吸引树，用于编码重要的代数和动力学不变量。其次，本项目将研究自由群和曲面群的扩张的双曲性。在过去的结果的基础上，PI将表征自由群扩展是双曲的，并将研究随机构造的自由群扩展的代数特征。通过与Kleinian群的类比，该项目还将发展一种新的映射类群的几何有限子群理论，并将其与萌芽中的分层双曲理论联系起来。第三，该项目将通过量化映射类组某些类型元素的普遍性和研究测地线和horocycle流的收缩目标特性，研究曲面上双曲结构的模空间。最后，在面向更一般的表面，该项目将探讨代数刚性方面的映射类组的无限型表面和动态刚性的长度谱的台球式动力系统。这个奖项反映了NSF的法定使命，并已被认为是值得支持的评估使用基金会的智力价值和更广泛的影响审查标准。

Award: DMS 2005368, Principal Investigator: Spencer D. DowdallThis project concerns two foundational branches of mathematics: group theory, and low-dimensional topology/geometry. A group is an algebraic framework that encodes symmetries of objects, such as the rigid motions of space, the configurations of a Rubik's cube, or the symmetries of a molecule. Low-dimensional topology and geometry concern the structure of space itself, like the surface of the earth or the universe we live in, together with inherent geometric features like curvature, distance, and volume. One way in which these topics are linked is through the concept of a parameter space (such as the geometric structures supported on a given object, or the configurations of points in that object) and its symmetry group. This project will focus on two important classes of examples in these contexts: group extensions, which are ways of building new groups out of old, and mapping class groups of surfaces, which are the symmetry groups for parameter spaces of hyperbolic structures on a surface. The purpose of the project is to identify and study key structural features, particularly rigid features that are invariant under perturbation or reparametrization, as well as general features that emerge from random constructions. This project provides research training opportunities for graduate students.Specifically, the project will investigate geometric and dynamical aspects of free group and surface automorphisms and explore how these interact with associated group extensions and moduli spaces of geometric structures. Firstly, the project will study cyclic extensions of surface groups and free groups. This entails the use of veering triangulations to study low-dilatation pseudo-Anosov surface homeomorphims, and the development of a parallel theory of veering triangulations for free-by-cyclic groups that will illuminate features of free group automorphisms. The project will also introduce a new universal attracting tree for free-by-cyclic groups that will be used to encode important algebraic and dynamical invariants. Secondly, the project will study hyperbolicity for extensions of free groups and surface groups. Building on past results, the PI will characterize when free group extensions are hyperbolic, and will study algebraic features of randomly constructed free group extensions. By means of analogy with Kleinian groups, the project will also develop a new theory of geometrically finite subgroups of mapping class groups and connect this to the budding theory of hierarchical hyperbolicity. Thirdly, the project will investigate the moduli space of hyperbolic structures on a surface by quantifying the prevalence of certain types of elements of the mapping class group and by studying shrinking target properties of the geodesic and horocycle flow. Finally, in a view towards more general surfaces, the project will explore aspects of algebraic rigidity in mapping class groups of infinite-type surface and dynamical rigidity for length spectra of billiard-style dynamical systems.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.