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Boundaries of Hyperbolic and Relatively Hyperbolic Groups

双曲群和相对双曲群的边界

基本信息

批准号：
1709964
负责人：
Genevieve Walsh
金额：
$ 19.57万
依托单位：
Tufts University
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2017
资助国家：
美国
起止时间：
2017-07-01 至 2021-06-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1709964&HistoricalAwards=false
关键词：
Boundaries Hyperbolic Relatively Groups

项目摘要

The concept of a group appears in several areas of pure and applied mathematics. Often a group has a topological space, its "boundary at infinity", that gives us some information about the group. The study of this brings the fields of topology, algebra, and dynamics into play. This project addresses the broad question of how much the boundary of a group determines the group. The principal investigator continually seeks to spread mathematical knowledge to a wide audience, including under-represented groups in mathematics and people working in industry and technology. This is done through speaking at and organizing conferences in a wide variety of venues aimed at wide audiences, and encouraging people in these groups (including industry) to attend her classes. In addition, all of the PI's mathematical results and findings are freely available to the public. The project addresses important and fundamental problems involving which hyperbolic or relatively hyperbolic groups can have certain boundaries. For one part of the project the principal investigator plans to extend known results about hyperbolic groups with planar or two-sphere boundaries to the relatively hyperbolic case, strengthening the connection with Kleinian groups and the fundamental groups of 3-manifolds. The proof methods sometimes are similar to the case of hyperbolic groups but often require different techniques. For the other part of the project she is exploring the wealth of planar boundaries than can occur as the complement of round disks in the 2-sphere.

群的概念出现在纯数学和应用数学的几个领域。通常一个群有一个拓扑空间，它的“无穷大边界”，这给了我们一些关于这个群的信息。对这一点的研究涉及到拓扑学、代数学和动力学领域。这个项目解决了一个群体的边界在多大程度上决定了这个群体的广泛问题。首席研究员不断寻求传播数学知识，以广泛的受众，包括在数学和在工业和技术工作的人代表性不足的群体。这是通过在各种各样的场所演讲和组织会议来实现的，目的是广泛的受众，并鼓励这些群体（包括行业）的人参加她的课程。此外，PI的所有数学结果和发现都免费向公众提供。该项目解决了重要的和基本的问题，涉及双曲或相对双曲群可以有一定的边界。对于该项目的一部分，首席研究员计划将已知的关于双曲群的结果与平面或两个球边界扩展到相对双曲的情况，加强与Kleinian群和3-流形的基本群的联系。证明方法有时类似于双曲群的情况，但通常需要不同的技术。对于该项目的另一部分，她正在探索丰富的平面边界，而不是作为2球中圆盘的补充。