Mathematical Sciences: Low Dimensional Topology and InfiniteGroup Theory

数学科学：低维拓扑和无穷群理论

• 批准号: 9001392
• 负责人: Peter Shalen
• 金额: $ 21.27万
• 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: 1990
• 资助国家: 美国
• 起止时间: 1990-06-01 至 1994-11-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9001392&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Low; Dimensional; Topology

Professors Culler and Shalen will continue their program for estimating Margulis constants, injectivity radii, and volumes of hyperbolic manifolds by using Sullivan-Patterson measures associated to 2-generator Kleinian groups. They will investigate group actions on generalized trees, extending research of Culler- Vogtmann and Gillet-Shalen. They will also apply the theory of character varieties to the study of Heegaard splittings of non- Haken 3-manifolds. Culler will continue his work on cohomological properties of the outer automorphism group of the free group. Shalen will study questions about amalgamated free product decompositions and deficiencies of subgroups, both in 3-manifold groups and in more general finitely presented groups. He will also investigate possible generalizations of Casson's invariant and connections with SL(2,C) representations of 3-manifold groups. These are all very active topics in the topology of 3- dimensional manifolds, i.e. very natural geometric objects of the same dimension as the space in which we live.

Culler和Shalen教授将继续他们的计划，通过使用与2-生成Klein群相关的Sullivan-Patterson度量来估计双曲流形的Marguis常数、内射性半径和体积。他们将研究广义树上的群作用，扩展了Culler-Vogtmann和Gillet-Shalen的研究。他们还将把特征标簇理论应用于非Haken 3-流形的Heegaard分裂的研究。卡勒将继续研究自由群的外自同构群的上同调性质。Shalen将在3-流形群和更一般的有限表示群中研究关于合并自由积分解和子群的亏性的问题。他还将研究Casson不变量的可能推广以及与3-流形群的SL(2，C)表示的联系。这些都是三维流形拓扑学中非常活跃的话题，即与我们生活的空间具有相同维度的非常自然的几何对象。