Boundaries of Hyperbolic and Relatively Hyperbolic Groups

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“无穷边界”，给了我们一些关于这个群的信息。对此的研究使拓扑学、代数学和动力学领域发挥了作用。这个项目解决了一个广泛的问题，即一个群体的边界在多大程度上决定了这个群体。首席研究员不断寻求将数学知识传播给更广泛的受众，包括在数学领域代表性不足的群体以及在工业和技术领域工作的人。这是通过在各种场所发表演讲并组织针对广泛受众的会议来实现的，并鼓励这些群体(包括行业)的人参加她的课程。此外，PI的所有数学结果和调查结果都可以免费向公众提供。该项目解决了双曲群或相对双曲群可以有一定边界的重要和根本问题。对于项目的一部分，主要研究者计划将关于具有平面或两个球面边界的双曲群的已知结果推广到相对双曲的情况，以加强与Klein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。对于项目的另一部分，她正在探索平面边界的丰富，而不是作为2-球体中圆形圆盘的补充。

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.