Subgroups of Mapping Class Groups

曲面S的映射类群Mod(S)是S的保定向同胚群，是几何群论和几何拓扑学中的一个基本研究对象。例如，映射类群与算术群(例如SL(n，Z))、自由群的自同构群和Artin群(例如辫子群)密切相关，使得Mod(S)作为更一般群论结果的应用场所和灵感两者都是有价值的。映射类群在3-和4-流形拓扑中也很重要，例如，通过Heegaard分裂和Lefschetz纤颤。此外，Mod(S)作为黎曼曲面的模空间的基本群而自然产生，因此在复分析和代数几何中得到了广泛的研究。PI有一个正在进行的研究MOD(S)的代数结构和理解MOD(S)、算术群和自由群的自同构群之间的关系的计划，方法是比较性质，如对组合模型的作用、自同构群和抽象公量群、生成集、有限性质、子群和可能的线性障碍。在提出的项目过程中，PI期望为两个群找到新的生成器，这两个群通过Heegaard分裂在3-流形的代数刻画中发挥重要作用：Torelli群和Heegaard群。PI还期望给出将Mod(S)与Coxeter群联系起来的Mod(S)的新表示，找到Hilden群的有限表示，并使用同调3-球面的Rochlin不变量产生的映射来洞察Torelli群和Johnson核的同调。理解约翰逊核的结构是特别有趣的，例如，人们可以使用这个群来构造所有同调的3-球。曲面是数学、物理和其他科学中的基本对象。近一个世纪以来，数学家们已经了解了如何对二维表面进行分类。然而，研究曲面自同构(曲面到自身的映射，保持曲面的本质属性)的自然第二步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)，已经得到了广泛的研究，但关于它的结构的一些最基本和本质的问题仍然没有解决。在这个项目中，首席研究员将继续一个正在进行的项目，研究MOD的代数结构(S)。虽然Mod(S)自然地出现在许多不同的数学领域，但这个项目的一个特殊目标是在Mod(S)中寻找揭示几何、拓扑和代数之间联系的代数结构。例如，一个重点将是研究Mod(S)的子群，它在各种三维空间的代数刻画中发挥着重要作用。另一个重点将是理解某些生成器集，或MOD(S)的“构建块”，它们揭示了曲面和反射和其他对称之间的关系，曲面是拓扑对象，反射和其他对称本质上是纯几何的。

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.