Intrinsic rigid structure in groups and surfaces

奖项：DMS 2005368，首席研究员：Spencer D.Dowdall这个项目涉及数学的两个基本分支：群论和低维拓扑/几何。群是编码对象对称性的代数框架，例如空间的刚性运动、魔方的构型或分子的对称性。低维拓扑和几何涉及空间本身的结构，如地球表面或我们生活的宇宙，以及固有的几何特征，如曲率、距离和体积。链接这些主题的一种方式是通过参数空间的概念(例如，给定对象上支持的几何结构，或该对象中的点的配置)及其对称组。这个项目将集中在这些上下文中的两类重要的例子：群扩张，它是从旧的群建立新的群的方法，以及映射曲面的类群，它是曲面上双曲结构的参数空间的对称群。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，以及随机构造产生的一般特征。这个项目为研究生提供了研究培训的机会。具体地说，该项目将研究自由群和曲面自同构的几何和动力学方面，并探索它们如何与几何结构的相关群扩张和模空间相互作用。首先，该项目将研究表面群和自由群的循环扩张。这需要使用转向三角剖分来研究低膨胀伪Anosov曲面同胚，并为自由循环群发展一个平行的转向三角剖分理论，它将阐明自由群自同构的特征。该项目还将为自由循环群引入一种新的通用吸引树，该树将用于编码重要的代数和动力学不变量。其次，该项目将研究自由群和表面群的扩张的双曲性。在过去结果的基础上，PI将刻画自由群扩张何时是双曲的，并将研究随机构造的自由群扩张的代数特征。通过与Klein群的类比，该项目还将发展映射类群的几何有限子群的新理论，并将其与萌芽中的分层双曲性理论联系起来。第三，该项目将通过量化映射类组中某些类型的元素的盛行率以及通过研究测地线和周期流的收缩目标属性来研究表面上双曲线结构的模空间。最后，着眼于更一般的表面，该项目将探索无限类型表面的映射类群的代数刚性和台球式动力系统长度谱的动态刚性的方面。这一奖项反映了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。

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.