Mathematical Sciences: Limit Sets of Kleinian Groups and Hyperbolic Groups

Bonahon将研究Kleinian群和双曲群的极限集。Kleinian群是双曲三维空间等距的离散群。Bonahon打算研究Kleinian群的极限集的拓扑类型，并证明它是由群的代数结构和相关商空间的某些几何不变量决定的。他还想研究双曲3-空间的叶在Kleinian群下不变的叶的渐近行为（即双曲3-流形的叶的提升），并分析这些叶接近群的极限集的速度。根据Gromov的术语，双曲群是一种有限生成的群，它具有类似于紧Kleinian群的某些生长性质。这样的双曲群有一个定义良好的极限集，也称为它的边界。Bonahon想要分析双曲群的极限集的某些拓扑性质与群的代数性质之间的关系，特别是它的外部自同构群和分解成混合积。在瑟斯顿的工作之后，近年来三维流形研究的进展相当令人惊讶地显示了双曲几何与理解三维流形结构的极端相关性。

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.