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

题目：CAT(0)群在边界上的作用摘要：几何群论的基本思想是把群G看作几何的刚性运动集合来研究无穷群G的结构。这样，群论问题就转化为几何问题，可以利用所有的几何技术。几何中的一个标准技术是通过紧化空间来研究空间的渐近性，即在空间上放置一个边界。单词双曲群是一类被证明非常有用的群。这是一类通过在几何上假设负曲率条件而得到的群。将这一发展良好的词双曲群理论扩展到“非正曲线”设置是目前的兴趣。已经提出了几种非正弯曲群。其中一类是由由一系列CAT(0)空间的刚性运动组成的群组成的，这些群在这里被称为“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提议的主要兴趣对象。对于字双曲群，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可以推广到非正曲线的情况这就是这里的观点。众所周知，用CAT(0)代替“双曲”这个词在许多定理中是行不通的，但找到正确的定理是统一非正弯曲群理论的重要一步，就像统一非正弯曲流形理论一样。CAT(0)群在空间边界上的作用动力学几乎没有被用来从几何群论的角度来回答任何问题。关于群体和群体的子群体的问题应该转化为关于这个动作的问题，就像在单词双曲设置中一样。群是数学中最基本的对象之一。加法下的整数集合就是大家都熟悉的群的一个例子。同样，一个几何物体的刚性运动的集合形成了一个群，事实上，这个群的数学性质通常表征了这个几何物体。例如，结是一种几何物体，最近在DNA研究中得到了很好的应用，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组，它完全决定了结。这些结群就是我们感兴趣的类型。

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.