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

这个项目涉及映射类群，它是曲面的同胚类的群。这里特别强调Torelli群，这个子群由平凡地作用于曲面同调的元素组成。我们采取三管齐下的方法，研究映射类群的同调、动力学和群论性质。这些技巧包括组合莫尔斯理论、泰希穆勒理论和几何群论的方法。在与Bestvina和Bux的共同工作中，我们的目的是确定Torelli群的同调有限性质；特别是回答他们是否有限表示的著名问题。与Farb和Leininger一起，我们致力于理解所有低熵伪Anosov的结构；特别是解决了对称性猜想，该猜想断言所有这样的伪Anosov都是作为旋转和支撑在一致有界亏格的次曲面上的映射的乘积得到的。利用Brendle，我们研究了超椭圆Torelli群的群结构，特别是Hain的猜想，该猜想指出群是由Dehn扭曲生成的，本课题的重点是映射类群，它是曲面(如球面或环面)的对称性的集合。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，我在几十年前关于映射班级组的全球结构的问题上取得了进展，这个项目致力于回答一些最重要的剩余问题。这项研究涉及的技术范围很广，从组合到代数，再到拓扑学，再到动力学，所提出的工作适用于代数几何，3-流形和群论。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.