Subgroups of Mapping Class Groups

映射类组的子组

• 批准号: 0504208
• 负责人: Tara Brendle
• 金额: --
• 依托单位: Cornell University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2005
• 资助国家: 美国
• 起止时间: 2005-07-01 至 2006-01-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0504208&HistoricalAwards=false
• 关键词: Subgroups; Mapping; Class; Groups

The mapping class group Mod(S) of a surface S is the group of orientation-preserving homeomorphisms of S, up to isotopy, and is a fundamental object of study in several fields, particularly in geometric group theory and geometric topology. For example, mapping class groups are closely related to arithmetic groups (e.g., SL(n,Z)), automorphism groups of a free group, and Artin groups (e.g., braid groups), making Mod(S) valuable both as a venue for applications of, and as an inspiration for, more general group theoretic results. Mapping class groups are also prominent in 3- and 4-manifold topology, e.g., via Heegaard splittings and Lefschetz fibrations. Moreover, Mod(S) arises naturally as the fundamental group of the moduli space of Riemann surfaces and is therefore much studied in complex analysis and algebraic geometry. The PI has an ongoing program for studying the algebraic structure of Mod(S) and understanding the relationship between Mod(S), arithmetic groups, and the automorphism group of a free group, by comparing properties such as actions on combinatorial models, automorphism and abstract commensurator groups, generating sets, finiteness properties, subgroups, and possible obstructions to linearity. In the course of the proposed project, the PI expects to find new generators for two groups which play an important role in the algebraic characterization of 3-manifolds via Heegaard splittings: the Torelli and Heegaard groups. The PI also expects to give a new presentation for Mod(S) relating Mod(S) to Coxeter groups, to find a finite presentation for the Hilden group, and to gain insight into the homology of the Torelli group and the Johnson kernel using a map arising from the Rochlin invariant of homology 3-spheres. Understanding the structure of the Johnson kernel is of particular interest as one can, for example, use this group to construct all homology 3-spheres.Surfaces are fundamental objects in mathematics, physics and other sciences. Mathematicians have understood how to classify 2-dimensional surfaces for nearly a century. However, the natural second step of investigating surface automorphisms (maps of a surface to itself which preserve the essential properties of the surface) has proved to be a much more challenging problem. The group of surface automorphisms, known as the mapping class group Mod(S) of the surface S, has been extensively studied, but some of the most basic and essential questions about its structure remain unsolved. In this project, the Principal Investigator will continue an ongoing program to study the algebraic structure of Mod(S). Though Mod(S) arises naturally in many different fields of mathematics, a particular goal of this project is to look for algebraic structure in Mod(S) which reveals connections between geometry, topology, and algebra. For example, one focus will be on the study of subgroups of Mod(S) which play a large role in the algebraic characterization of various 3-dimensional spaces. Another focus will be understanding certain sets of generators, or ``building blocks'' of Mod(S), which reveal the relationship between surfaces, which are topological objects, and reflections and other symmetries, which are purely geometric in nature.

曲面S的映射类群Mod（S）是S的保向同胚群，直到合痕，并且是几个领域的基本研究对象，特别是在几何群论和几何拓扑学中。 例如，映射类组与算术组密切相关（例如，SL（n，Z））、自由群的自同构群和Artin群（例如，辫子群），使模（S）作为一个场所的应用程序，并作为一个灵感，更一般的群论结果的价值。 映射类群在3-和4-流形拓扑中也很突出，例如，通过Heegaard分裂和Lefschetz纤维化。此外，Mod（S）作为黎曼曲面的模空间的基本群自然出现，因此在复分析和代数几何中得到了大量研究。PI有一个正在进行的项目，用于研究Mod（S）的代数结构，并通过比较诸如组合模型上的作用、自同构和抽象分解群、生成集、有限性性质、子群和可能的线性障碍等性质来理解Mod（S）、算术群和自由群的自同构群之间的关系。 在拟议项目的过程中，PI希望找到两个群的新生成元，这两个群在通过Heegaard分裂的三维流形的代数特征中发挥重要作用： Torelli和Heegaard集团 PI还期望给出一个新的表示，Mod（S）与Coxeter群的关系，找到希尔登群的有限表示，并使用同源3-球面的Rochlin不变量映射深入了解Torelli群和约翰逊核的同源性。 理解约翰逊核的结构是特别有意义的，因为人们可以，例如，使用这个群来构造所有的同调3-sphere。曲面是数学，物理和其他科学中的基本对象。 近世纪以来，数学家们已经了解了如何对二维表面进行分类。 然而，自然的第二步，调查表面自同构（映射的表面本身保持的基本性质的表面）已被证明是一个更具挑战性的问题。 曲面自同构群，称为曲面S的映射类群Mod（S），已经被广泛研究，但关于它的结构的一些最基本和最本质的问题仍然没有解决。 在这个项目中，主要研究者将继续进行一个正在进行的计划，以研究Mod（S）的代数结构。 虽然Mod（S）在许多不同的数学领域中自然出现，但这个项目的一个特定目标是寻找Mod（S）中的代数结构，揭示几何，拓扑和代数之间的联系。例如，一个重点将是研究子群的模（S）发挥了很大的作用，在代数表征的各种三维空间。 另一个重点将是理解某些生成器集，或Mod（S）的“构建块”，它揭示了表面（拓扑对象）与反射和其他对称性（本质上纯粹是几何的）之间的关系。