Mathematical Sciences: Limit Sets of Kleinian Groups and Hyperbolic Groups

数学科学：克莱因群和双曲群的极限集

• 批准号: 9001895
• 负责人: Francis Bonahon
• 金额: $ 5.73万
• 依托单位: University of Southern California
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1990
• 资助国家: 美国
• 起止时间: 1990-06-01 至 1992-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9001895&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Limit; Sets; Kleinian

Bonahon will study limit sets of Kleinian groups and of hyperbolic groups. A Kleinian group is a discrete group of isometries of hyperbolic 3-space. Bonahon intends to study the topological type of the limit set of a Kleinian group and show that it is determined by the algebraic structure of the group together with certain geometric invariants of the associated quotient space. He also wants to study the asymptotic behavior of the leaves of foliations of hyperbolic 3-space which are invariant under a Kleinian group (namely lifts of foliations of hyperbolic 3-manifolds) and to analyze how fast these leaves approach the limit set of the group. Following Gromov's terminology, a hyperbolic group is a finitely generated group which has certain growth properties similar to those of a cocompact Kleinian group. Such a hyperbolic group has a well defined limit set, also called its boundary. Bonahon wants to analyze how certain topological properties of the limit set of a hyperbolic group are related to algebraic properties of the group, involving in particular its outer automorphism group and decompositions into amalgamated products. Advances in the study of 3-dimensional manifolds in recent years, following work of Thurston, has rather surprisingly shown the extreme relevance of hyperbolic geometry to the understanding of the structure of 3-manifolds.

Bonahon将研究极限集的Kleinian群和 双曲群 Kleinian群是一个离散群， 三维双曲空间的等距 Bonahon打算研究 Kleinian群极限集的拓扑型，并证明了 它是由群的代数结构决定的 以及相关的几何不变量 商空间 他还想研究 双曲3-空间的叶理的叶， 在Kleinian群下不变（即 双曲三维流形），并分析如何快速这些叶 接近该组的极限集。 根据Gromov的术语，双曲群是一个 具有一定增长性质的非线性生成群 类似于上紧Kleinian群。 这样的 双曲群有一个定义良好的极限集，也称为其 边界 Bonahon想分析某些拓扑结构 一个双曲群的极限集的性质与 群的代数性质，特别是它的 外自同构群和分解成合并的 产品. 近年来三维流形研究的进展 在瑟斯顿的研究之后，令人惊讶的是， 双曲几何对于理解 三维流形的结构