CAREER: GROUP-THEORETICAL, DYNAMICAL, AND COMBINATORIAL ASPECTS OF MAPPING CLASS GROUPS

这个项目涉及映射类组，这是一个曲面同胚的合痕类的组。 特别强调的是Torelli群，该子群由对表面的同源性起平凡作用的元素组成。 我们采取了三管齐下的方法，对同调，动力学和群论性质的映射类群。 这些技术包括组合莫尔斯理论、Teichmuller理论和几何群论的方法。 在联合工作与Besteland和布克斯，我们的目标是确定同调有限性质的Torelli群，特别是要回答著名的问题，他们是否是considered提出。 与Farb和Leininger，我们的工作理解所有低熵伪Anosovs的结构，特别是解决对称猜想，它断言，所有这些伪Anosovs获得的产品的旋转与地图上支持的地下一致有界的属。 利用Brendle，我们研究了超椭圆Torelli群的群结构，特别是海恩的猜想，该猜想指出该群是由Dehn扭曲生成的。这个项目的重点是映射类群，它是 曲面的对称性的集合，如球面或环面。这是数学中的一个中心对象，与代数几何，双曲几何和动力学有关。 这一领域的发现对数学以外的领域产生了影响，如弦论和密码学。 与我的合作者一起，我在关于映射类组的全球结构的几十年前的问题上取得了进展，这个项目致力于回答一些最重要的遗留问题。 在这项研究中所涉及的技术是广泛的，从组合，代数，拓扑，动力学，和拟议的工作有应用代数几何，3-流形，群论。 我将与几位合作者和不同层次的学生一起研究该提案的研究方面。 此外，我还将为拓扑学的年轻研究人员举办一个研讨会，目的是培训和培养未来的研究人员和教育工作者。

This project concerns the mapping class group, which is the group of isotopy classes of homeomorphisms of a surface. There is particular emphasis on the Torelli group, the subgroup consisting of elements that act trivially on the homology of the surface. We take a three- pronged approach, on the homological, dynamical, and group-theoretical properties of mapping class groups. The techniques include methods from combinatorial Morse theory, Teichmuller theory, and geometric group theory. In joint work with Bestvina and Bux, we aim to determine the homological finiteness properties of the Torelli groups; in particular to answer the famous question of whether or not they are finitely presented. With Farb and Leininger, we work on understanding the structure of all low entropy pseudo-Anosovs; in particular to address the Symmetry Conjecture, which asserts that all such pseudo- Anosovs are obtained as the product of a rotation with a map supported on a subsurface of uniformly bounded genus. With Brendle, we study the group structure of the hyperelliptic Torelli group, in particular focused on the conjecture of Hain, which states that the group is generated by Dehn twists.The focus of this project is the mapping class group, which is the collection of symmetries of a surface, such as a sphere or a torus. This is a central object in mathematics, with connections to algebraic geometry, hyperbolic geometry, and dynamics. Discoveries in this area have had influence on fields outside of mathematics, such as string theory and cryptography. With my collaborators, I have made progress on decades-old questions about the global structure of mapping class group, and this project is dedicated to answering some of the most important remaining questions. The techniques involved in this study are wide-ranging, from combinatorial, to algebraic, to topological, to dynamical, and the proposed work has applications to algebraic geometry, 3-manifolds, and group theory. I will work on the research aspects of the proposal with several collaborators and with students at various levels. In addition, I will host a workshop for young researchers in topology, with the aim of training and cultivating future researchers and educators.