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作用来实现的。几何群理论家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，可以自然地产生一组方程或参数。因此，几何群论的研究是至关重要的。在这个建议中，我们感兴趣的群体作用于树，一个理论，是由巴斯和塞尔发起的，并有深远的影响分裂的群体合并的自由产品和HNN扩展。如果一个组分裂，那么可以将其分解为更小，更易于管理的部分。基本群将群论与几何学和拓扑学联系起来。在某些情况下，分裂流形的基本群是将流形本身分裂成更小更好理解的子流形的关键。这与虚哈肯猜想等数学中的重要命题有关。我们希望这项工作所产生的建议将启发我们的现象分裂的群体和流形。该项目的第二个主题是展示我们称之为渐近CAT（0）空间的类的丰富性，这些空间是度量空间，当从越来越远的观测点观察时，似乎具有非正曲率。树和双曲空间是自然的例子。我们希望构造出在Gromov意义下不是双曲的渐近CAT（0）图的例子。这与计算机科学家和图形理论家所推崇的鄂尔多斯问题有关。显然，解决鄂尔多斯问题将使我们能够解决几何问题的算法在计算机上。

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.