Mathematical Sciences: Trees, 3-Orbifolds, and Hyperbolic Groups

这个方案的问题在于重叠区域 低维拓扑与组合群 理论 W.瑟斯顿的研究表明， 3-流形是双曲的，即允许黎曼流形的流形 常负曲率度量 通过让一个序列 这样的结构退化，得到树。 第一 问题是要了解一个群体的行动空间， 树 这里的动机问题是， 单形作用空间在所有单形作用空间中是稠密的。 行动 第二个问题是限制 3-用它们的基本群来表示。 主 这里的问题是是否有一个界限（就 组所需的发电机数量）， 极大有限子群的共轭类 由于M. Gromov，任意 基本群双曲性概念的群 一个封闭的流形 最后一个问题是要表明， 不可约映射柱的基本群 自由群的自同构在这个意义上是双曲的。 所有的问题都有一个共同点， 拓扑学和数学、几何学或其他分支 代数，或者两者都有。 这是一个常年存在的现象， 数学，但它的盛衰。 目前，项目 形成相邻场之间的桥梁， 增殖 模糊界限的趋势，就像在这个 项目，尤其突出。

The problems in this program are in the area of overlap between low-dimensional topology and combinatorial group theory. W. Thurston's work indicates that the most interesting 3-manifolds are hyperbolic, i.e. ones that admit a Riemannian metric of constant negative curvature. By letting a sequence of such structures degenerate, a tree is obtained. The first problem is to understand the space of actions of a group on trees. The motivating question here is whether or not the space of simplicial actions is dense in the space of all actions. The second problem is to bound the complexity of 3-orbifolds in terms of their fundamental groups. A main question here is whether there is a bound (in terms of the number of generators needed for the group) for the number of conjugacy classes of maximal finite subgroups. There is a generalization, due to M. Gromov, to arbitrary groups of the notion of hyperbolicity of the fundamental group of a closed manifold. The final problem is to show that the fundamental group of a mapping cylinder of an irreducible automorphism of a free group is hyperbolic in this sense. All the problems have in common that they heavily mix topology and some other branch of mathematics, geometry or algebra, or both. This is a perennial phenomenom in mathematics, but it waxes and wanes. At the moment projects which form bridges between neighboring fields are proliferating. The trend to blur boundaries, as in this project, is particularly pronounced.