Boundaries of Nonpositively Curved Groups

非正弯曲群的边界

• 批准号: 9704939
• 负责人: Kim Ruane
• 金额: $ 4万
• 依托单位: Vanderbilt University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1997
• 资助国家: 美国
• 起止时间: 1997-08-01 至 1999-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9704939&HistoricalAwards=false
• 关键词: Boundaries; Nonpositively; Curved; Groups

9704939 Ruane An important class of groups for which geometric ideas have proven useful is the class of word hyperbolic groups proposed by M. Gromov. These are groups which discretely approximate a geometry more like that of the hyperbolic plane than the Euclidean plane. It is currently of interest to extend this well-developed theory of word hyperbolic groups to the ``nonpositively curved'' setting. Just as word hyperbolic groups are a generalization of the classical hyperbolic groups, finitely generated free groups, and certain small cancellation groups, there should be a general class of nonpositively curved groups that includes finitely generated free abelian groups, more general small cancellation groups, and fundamental groups of Riemannian manifolds of nonpositive curvature. Recently, there have been several proposed classes of nonpositively curved groups. One such class consists of groups that arise via geometric actions on ``CAT(0)'' spaces. These are spaces which enjoy many of the same geometric properties of universal covers of Riemannian manifolds of nonpositive curvature. Both the Euclidean and hyperbolic planes are examples of CAT(0) spaces. The boundary of a CAT(0) space which admits a geometric group action is an object of great interest in the area. Recently, several important problems in group theory have been solved with the use of geometric methods. For word hyperbolic groups, the boundary has proven a useful tool. Many of the theorems which 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 right theorems is an important step in unifying the theory of nonpositively curved groups, much like the theory of nonpositively curved manifolds. The basic idea of Geometric Group Theory is to study the structure of an infinite group G by studying ``geometric'' actions of G on different geometries. In this way, any such group is viewed as a set of rigid motions of a geometry. An example to keep in mind is that of the Euclidean plane. This is a geometry whose boundary can be identified with the unit circle, where each point of the circle represents a direction in the plane which heads out to infinity. A rigid motion in a straight line is known as a translation. This space (the plane) may be acted upon by translations in any direction, but translations in two independent directions will suffice to give all of them. Thus two (commuting) copies of the group of integers can be thought of as acting on the plane by translation. In fact, this geometric setup determines this group almost uniquely. In general, when a group acts geometrically on a geometry, there is a ``picture'' of the group inside the space created by following the image of one point under all of the group elements. Then the group, which started out as an abstract mathematical object, can be studied by studying the geometry of the space, where the problems are now geometric instead of algebraic. In the example above, the group consists of moves from one square to another on an infinite chess board. The collection of moves may then be given a concrete geometric picture by indentifying each square with its center, the totality of moves becoming a lattice that discretely approximates the plane. ***

9704939 RUANE一个重要的小组被证明有用的几个小组是M. Gromov提出的一类单词双曲线组。 这些基团离散地近似于比几何平面的几何形状，而不是欧几里得平面。 目前，将这种发达的单词双曲线群体的理论扩展到``非主体弯曲''的设置是令人感兴趣的。 正如单词双曲线群是经典双曲线组的概括，有限生成的自由组和某些小型取消组一样，应该有一类的非稳态弯曲组，其中包括有限生成的自由的Abelian群体，更一般的小型取消组，以及无稳定策划的Riemannian份谱的基本组。 最近，已经有几类非阳性弯曲的组类。 这样的类是通过在``cat（0）'空格上通过几何作用出现的群体组成的。 这些空间享有非阳性曲率的Riemannian流形的普遍覆盖物的许多几何特性。 欧几里得和双曲线平面都是CAT（0）空间的示例。 承认几何群体动作的猫（0）空间的边界是该地区引起极大兴趣的对象。 最近，使用几何方法解决了小组理论中的几个重要问题。 对于单词双曲线组，边界已证明是有用的工具。 在该设置中持有的许多定理都应对非物质弯曲的设置有所概括，这就是这里采取的观点。 众所周知，在许多这些定理中，用``cat（0）''代替``单词双曲线''''''''''单词''''''''''''中的许多定理都行不通，但是找到正确的定理是统一非积极弯曲群体理论的重要一步，这与非积极性弯曲的流膜理论一样。 几何群体理论的基本思想是通过研究G对不同几何形状的``几何''作用来研究无限G组的结构。 这样，任何此类群体都被视为几何形状的一组刚性动作。 要记住的一个例子是欧几里得平面。 这是一个几何形状，可以用单位圆来识别边界，其中圆的每个点代表平面中的一个方向，该方向朝着无穷大。 直线上的刚性运动称为翻译。 这个空间（平面）可以通过任何方向的翻译来实现，但是在两个独立方向上的翻译足以给予所有方向。 因此，可以将整个整数的两个（通勤）副本视为通过翻译在平面上作用。 实际上，这种几何设置几乎决定了这一组。 通常，当一个组在几何形状上几何作用时，通过在所有组元素下遵循一个点的图像来创建的空间内部的组``图片''。 然后，可以通过研究空间的几何形状来研究该组作为抽象数学对象的开始，该空间现在是几何而不是代数。 在上面的示例中，该小组包括在无限的国际象棋板上从一个正方形到另一个正方形的移动。 然后，可以通过将每个正方形的中心缩进每个正方形来给出一个具体的几何形状图片，整数变成了一个晶格，可离散地近似平面。 ***