The Action of a CAT(0) Group on the Boundary

建议：DMS-9973119 PI：Kium Ruane题目：CAT（0）群在边界上的作用摘要：几何群论的基本思想是通过将G看作几何的某个刚性运动集合来研究无限群G的结构. 这样，群论问题就转化为几何问题，所有的几何技巧都可以利用。 几何学中的一个标准技术是通过紧致化空间来研究空间的渐近性-即在空间上放置边界。 字双曲群是一类群，这个想法已被证明是非常有用的。 这是一类通过在几何上假设负曲率条件而得到的群。 它是目前的兴趣，以扩大这一良好的发展理论的字双曲群的“nonpositively curved”的设置。 有几类非正曲群被提出。 一类这样的群由CAT（0）空间的一组刚性运动所产生的群组成，这里称之为“CAT（0）群”。 群G作用于CAT（0）空间的边界以及G在该边界上的作用将是本建议中感兴趣的主要对象。 对于双曲群，通过研究群的边界和群在边界上的作用，得到了许多关于群的定理。 许多在这种情况下成立的定理应该推广到非正曲线的情况，这就是这里所采用的观点。 我们已经知道，在许多定理中用CAT（0）代替“双曲”一词是行不通的，但是找到正确的定理是统一非正曲群理论的重要一步，就像非正曲流形理论一样。 CAT（0）群在空间边界上作用的动力学几乎没有被用来回答几何群论观点的任何问题。 关于群和群的子群的问题应该转化为关于这个作用的问题，就像在双曲设置这个词中一样。群是数学中最基本的对象之一。 加法下的整数集合是每个人都熟悉的群的一个例子。 此外，几何对象的刚性运动的集合形成一个组，并且事实上，该组的数学性质通常表征几何对象。 例如，结是几何物体，最近在DNA研究中看到了美丽的应用，因为我们体内的DNA链经常打结。 在这些应用中，能够区分两种不同的结是一个重要的想法。 有一个与结相关的自然群，它完全决定了这个结。 这些纽结群是我们在这里感兴趣的类型的群。

Proposal: DMS-9973119PI: Kium RuaneTitle: The Action of a CAT(0) Group on the BoundaryAbstract: The basic idea of Geometric Group Theory is to study the structure of an infinite group G by viewing G as a certain set of rigid motions of a geometry. In this way, group-theoretic problems are translated into geometric problems and all the techniques of geometry can then be utilized. A standard technique in geometry is to study the asymptotics of a space by compactifying it - i.e. putting a boundary on the space. Word hyperbolic groups are a class of groups for which this idea has proven to be incredibly useful. This is a class of groups obtained by assuming a negative curvature condition on the geometry. It is currently of interest to extend this well-developed theory of word hyperbolic groups to the ``nonpositively curved'' setting. There have been several proposed classes of nonpositively curved groups. One such class consists of groups that arise as a set of rigid motions of a CAT(0) space, groups referred to here as ``CAT(0) groups". The boundary of a CAT(0) space on which group G acts as well as the action of G on this boundary will be the main objects of interest in this proposal. For word hyperbolic groups, studying the boundary and the action of the group on the boundary has yielded many theorems about the group. Many of the theorems that hold in that setting should have generalizations to the nonpositively curved setting and that is the point of view taken here. It is already known that replacing the phrase "word hyperbolic" with CAT(0) in many of these theorems is not going to work, but finding the correct theorems is an important step in unifying the theory of nonpositively curved groups much like the theory of nonpositively curved manifolds. The dynamics of the action of a CAT(0)group on the boundary of the space has hardly been used to answer any questions from the geometric group theory point of view. Questions about the group and subgroups of the group should translate into questions about this action, much like in the word hyperbolic setting.A group is one of the most fundamental objects in mathematics. The set of whole numbers under addition is an example of a group that everyone is familiar with. Also, the set of rigid motions of a geometric object forms a group and in fact, the mathematical properties of this group often characterize the geometric object. For example, knots are geometric objects which have recently seen beautiful applications in the study of DNA since the strands of DNA in our body are often knotted. Being able to tell the difference between two different knots is an important idea in these applications. There is a natural group associated to a knot which completely determines the knot. These knot groups are groups of the type we are interested in here.