Mathematical Sciences: Low-dimensional topology and infinitegroups

数学科学：低维拓扑和无限群

• 批准号: 8701804
• 负责人: Peter Shalen
• 金额: $ 18.06万
• 依托单位: University of Illinois at Chicago
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1987
• 资助国家: 美国
• 起止时间: 1987-06-15 至 1990-11-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=8701804&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Low; dimensional; topology

Professor Shalen will continue his work with H. Gillet on the structure of groups acting on lambda-trees. They will then explore applications to hyperbolic geometry, K-theory and linear groups. Shalen will also continue his work with P. Wagreich on growth functions of groups and applications to the study of volume and diameters of hyperbolic manifolds, as well as his work with G. Baumslag and J. Morgan in which properties of finitely presented groups are derived from studying their varieties of linear representations over fields. Professors Culler and Shalen will continue their work on Dehn surgery on knots with the ultimate goal of classifying surgeries that give manifolds with cyclic fundamental groups and settling the Property P conjecture. They will also work on extending the Cyclic Surgery Theorem to links, and interpreting the new polynomial invariants for knots and links in terms of varieties of groups and representations. Professor Culler will continue his work with K. Vogtmann on the outer automorphism group of a free group. They will develop further the analogy between this group and arithmetic groups. Culler will also study the dynamics of these outer automorphisms in analogy with Thurston's work on dynamics of surface automorphisms. In addition, he will work on constructing generalized buildings for a class of goups that includes these outer antomorphism groups as well as arithmetic groups and mapping class groups. Exploring these numerous strong connections between low- dimensional topology and group theory will advance both fields, with ultimate impact on mathematical physics and other heavy users of the machinery of modern mathematics, particularly on users of precise and concise ways of describing the symmetries of highly connected spaces.

Shalen教授将继续他与H.Gillet在作用于Lambda-Tree的群结构方面的工作。然后，他们将探索在双曲几何、K理论和线性群中的应用。Shalen还将继续他与P.Wgreich在群的增长函数方面的工作以及在双曲流形的体积和直径研究中的应用，以及他与G.Baumgrag和J.Morgan的工作，在这些工作中，有限表示群的性质是通过研究域上它们的线性表示的变化而得到的。卡勒和沙伦教授将继续他们在纽结上的Dehn手术的工作，最终目标是对给出具有循环基本群的流形的手术进行分类，并解决性质P猜想。他们还将致力于将循环运算定理推广到环，并根据各种群和表示来解释结和环的新的多项式不变量。卡勒教授将继续他与K.Vogtmann关于自由群的外自同构群的工作。他们将进一步发展这个群和算术群之间的类比。卡勒还将研究这些外自同构的动力学，类似于瑟斯顿关于表面自同构动力学的工作。此外，他将致力于为一类群构造广义建筑，该群包括这些外反同构群以及算术群和映射类群。探索低维拓扑学和群论之间的众多强大联系将推动这两个领域的发展，最终将对数学物理学和其他大量使用现代数学机器的用户产生影响，特别是对使用精确和简洁的方法描述高度连接空间的对称性的用户。