Groups acting on Asymptotically CAT(0) spaces

几何群论是研究对称性的数学工具。这是通过研究群体在几何空间上的行为来实现的。几何群论者研究的一类基本对象是流形或簇，它们是方程集合的解集。在数学、科学和经济学的不同领域中，方程式或参数的集合可能自然而然地出现。因此，几何群论的研究是极其重要的。在这个提议中，我们对作用在树上的群感兴趣，这一理论是由Bass和Serre开创的，并对群的分裂产生了深远的影响，如合并的自由积和HNN扩张。如果一个群体分裂，那么人们可以将其分解成更小、更容易管理的部分。基本群将群论与几何学和拓扑学联系起来。在某些情况下，分割流形的基本群是将流形本身分割成更小、更容易理解的子流形的关键。这与数学中的重要猜想有关，比如虚拟哈肯猜想。我们希望，这项提议所产生的工作将启发我们注意集团分裂和多种形式分裂的现象。该项目的第二个主题是展示我们所称的渐近CAT(0)空间类的丰富性，这些空间启发式地是从越来越远的观察点观察时看起来具有非正曲率的度量空间。树和双曲空间是很自然的例子。我们希望构造在Gromov意义下非双曲的渐近CAT(0)图的例子。这与鄂尔多斯的一个问题有关，这个问题受到计算机科学家和图论学者的推崇。显然，鄂尔多斯问题的解决将使我们能够在计算机上用算法解决几何问题。

Geometric group theory is the mathematical tool for studying symmetry. This is achieved by investigating groups via their actions on geometric spaces. A fundamental class of objects studied by geometric group theorists are manifolds or varieties which are solutions sets to collections of equations. A collection of equations or parameters could arise naturally in different areas of mathematics, of science and in economics. Hence research in geometric group theory is of paramount importance. In this proposal we are interested in groups acting on trees, a theory that was initiated by Bass and Serre and has profound influence on the splittings of groups as amalgamated free products and HNN extensions. If a group splits, then one can decompose it into smaller, more manageable pieces. The fundamental group links group theory to geometry and topology. Splitting the fundamental group of a manifold holds the key in some cases to splitting the manifold itself into smaller better-understood sub-manifolds. This is related to important conjectures in mathematics like the virtual Haken conjecture. We hope that the work arising from this proposal will enlighten us on the phenomenon of splittings of groups and manifolds. A second theme of the project is to exhibit the richness of the class of what we call asymptotically CAT(0) spaces which heuristically are metric spaces appearing to have non-positive curvature when viewed from increasingly distant observation points. Trees and hyperbolic spaces are natural examples. We hope to construct examples of asymptotically CAT(0) graphs which not hyperbolic in the sense of Gromov. This is related to a question of Erdos held in esteem among computer scientists and graph theorists. Evidently a resolution of Erdos' question will allow us to solve geometric problems algorithmically on computer.