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Geometry and Dynamics in Surfaces and Free Group Extensions

曲面中的几何和动力学以及自由群扩展

基本信息

批准号：
1711089
负责人：
Spencer Dowdall
金额：
$ 16.99万
依托单位：
Vanderbilt University
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2017
资助国家：
美国
起止时间：
2017-06-01 至 2022-05-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1711089&HistoricalAwards=false
关键词：
Geometry Dynamics Surfaces Free Group

项目摘要

The algebraic properties of the symmetries of an object are encoded by the mathematical structure known as a group. Formally, a group is simply a set equipped with a binary operation that satisfies two axioms formalizing the concept of undoing the operation. Mathematicians are concerned both with the overarching abstract algebraic structures that arise from these rules, and with concrete examples that have natural significance, such as the set of rigid motions of the plane, the configurations of a Rubik's cube, or the symmetries of a polyhedron. Here free groups, which have a particularly simple algebraic structure in which there is the minimal possible cancellation, play an essential role as building blocks from which many other groups are built. Free group extensions are a basic example of this sort of construction. Yet, despite being built from well-understood pieces, free group extensions remain rather mysterious. This project will investigate dynamical, algebraic, and geometric aspects of free group extensions and combine these perspectives to illuminate the broad structure of these groups. This expands on the investigator's prior work with collaborators exploring and relating the different ways a given group may be built as a free-by-cyclic group, as well as work on the coarse negative curvature of certain free group extensions.This project will focus on aspects of geometry, topology, algebra, and dynamics in the context of free group automorphisms and surface homeomorphisms. The investigator will continue his work with collaborators analyzing splittings of free-by-cyclic groups via the dynamics of certain semi-flows on 2-complexes. A main goal is to tie together all the monodromies of the different splittings via a uniformized action of the entire group on a tree and to use this to define canonical algebraic invariants and relate all the Cannon-Thurston maps to each other. In addition to strengthening these connections, the project will work to prove that the property of having a fully irreducible monodromy is shared by either all or none of the splittings of the group. Continuing his work with a collaborator, the investigator will study the geometry of general free group extensions and work to characterize Gromov hyperbolicity and establish rigidity results for this class of groups. With regard to the study of surface homeomorphisms, the investigator will work with collaborators to show that all small-dilatation pseudo-Anosov homeomorphisms have the simple structure predicted by the symmetry conjecture. Finally, the investigator will study large-scale geometric and dynamical properties of the Teichmuller space of a surface with an aim towards calculating expected thinness of triangles and lattice point asymptotics for natural sets of surface homeomorphisms.

一个物体的对称性的代数性质由称为群的数学结构编码。形式上，一个群只是一个配备了二元运算的集合，它满足两个公理，形式化了撤销运算的概念。数学家既关心从这些规则中产生的总体抽象代数结构，也关心具有自然意义的具体例子，例如平面的刚性运动集，魔方的配置或多面体的对称性。在这里，自由群，其中有一个特别简单的代数结构，其中有最小的可能取消，发挥了重要的作用，作为积木，从许多其他群体的建设。自由群扩张是这种构造的一个基本例子。然而，尽管自由组扩展是由很好理解的部分构建的，但自由组扩展仍然相当神秘。这个项目将调查动力学，代数和几何方面的自由群扩展和联合收割机这些观点来阐明这些群体的广泛结构。这扩展了研究者以前的工作与合作者探索和相关的不同方式，一个给定的群体可能被建立为一个自由的循环群，以及工作的粗负曲率的某些自由群extensions.这个项目将集中在几何，拓扑，代数和动力学方面的自由群自同构和表面同胚的背景下。研究人员将继续他的工作与合作者分析分裂的自由循环组通过动力学的某些半流2-复合物。一个主要的目标是通过一个树上的整个群的一致化作用将不同分裂的所有单值性联系在一起，并使用这个来定义正则代数不变量，并将所有的Cannon-Thurston映射相互联系起来。除了加强这些联系之外，该项目还将努力证明具有完全不可约单值性的性质被群的所有分裂共享或不共享。继续他的工作与合作者，研究人员将研究几何的一般自由群扩展和工作的特点格罗莫夫双曲性和建立刚性的结果，这类群体。关于曲面同胚的研究，研究者将与合作者一起证明所有小膨胀伪Anosov同胚都具有对称猜想所预测的简单结构。最后，研究者将研究表面的Teichmuller空间的大规模几何和动力学性质，目的是计算表面同胚的自然集的三角形和格点渐近的预期薄度。