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

本程序中的问题是低维拓扑和组合群论之间的重叠区域。W·瑟斯顿的工作表明，最有趣的3-流形是双曲的，即允许常负曲率的黎曼度量的流形。通过让这种结构的序列退化，就得到了一棵树。第一个问题是理解一个群体在树上的动作空间。这里的激励问题是，在所有动作的空间中，单纯动作的空间是否稠密。第二个问题是用其基本群来界定3-orborold的复杂性。这里的一个主要问题是极大有限子群的共轭类的个数是否有界(根据群所需的生成元数)。由于M.Gromov，把闭流形的基本群的双曲性的概念推广到任意群。最后一个问题是证明自由群的不可约自同构的映射柱面的基本群在这个意义上是双曲的。所有这些问题都有一个共同点，那就是它们严重混合了拓扑学和其他一些数学、几何或代数的分支，或者两者兼而有之。这是数学中一个长期存在的现象，但它时好时坏。目前，在邻近油田之间架起桥梁的项目正在激增。模糊边界的趋势尤其明显，就像在这个项目中一样。

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.