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

这个程序的问题是在低维拓扑和组合群论之间的重叠领域。W. Thurston的工作表明，最有趣的3流形是双曲的，即那些承认恒定负曲率的黎曼度规的流形。通过让一个这样的结构序列退化，得到一个树。第一个问题是理解一个群体在树上的活动空间。这里的激励问题是，简单行动的空间在所有行动的空间中是否密集。第二个问题是根据它们的基本群来限定三轨道的复杂性。这里的一个主要问题是最大有限子群的共轭类的数量是否有一个界限（就群所需的生成器的数量而言）。由于M. Gromov的存在，我们将闭流形的基本群的双曲性的概念推广到任意群。最后的问题是在这个意义上证明自由群的不可约自同构的映射柱面的基群是双曲的。所有的问题都有一个共同点，那就是它们大量地混合了拓扑学和其他一些数学分支，几何或代数，或两者兼而有之。这是数学中一个长期存在的现象，但它有盛有衰。目前，在邻近油田之间形成桥梁的项目正在激增。在这个项目中，模糊界限的趋势尤为明显。

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.