Boundaries of Nonpositively Curved Groups

9704939 Ruane是一类重要的群，它的几何思想已被证明是有用的，这类群是由M.Gromov提出的单词双曲群。这些群离散地逼近一个更像双曲平面而不是欧几里得平面的几何。目前感兴趣的是将这一发展得很好的双曲群理论推广到“非正曲线”环境。就像字双曲群是经典双曲群、有限生成自由群和某些小消去群的推广一样，应该有一类一般的非正曲群，它包括有限生成的自由阿贝尔群、更一般的小消去群和非正曲率黎曼流形的基本群。最近，人们提出了几类非正曲群。一个这样的类由通过在‘`CAT(0)’空间上的几何作用而产生的群组成。这些空间具有许多与非正曲率黎曼流形的泛覆盖相同的几何性质。欧几里得平面和双曲平面都是CAT(0)空间的例子。允许几何群作用的CAT(0)空间的边界是该区域中非常感兴趣的对象。近年来，用几何方法解决了群论中的几个重要问题。对于字双曲群，边界已被证明是一个有用的工具。在这种情况下成立的许多定理应该具有对非正曲线设置的推广，这就是这里所采用的观点。人们已经知道，在许多这样的定理中，用“CAT(0)”替换短语“单词双曲线”是行不通的，但找到正确的定理是统一非正曲群理论的重要一步，就像非正曲线流形理论一样。几何群论的基本思想是通过研究G在不同几何上的“几何”作用来研究无限群G的结构。通过这种方式，任何这样的组都被视为几何体的一组刚性运动。一个需要牢记的例子是欧几里得平面。这是一种其边界可以用单位圆标识的几何图形，其中圆的每个点代表平面中朝向无穷远的方向。直线上的刚体运动称为平移。这个空间(平面)可以作用于任何方向的平移，但在两个独立方向上的平移就足以给出所有的平移。因此，这组整数的两个(通勤)副本可以被认为通过平移作用于平面上。事实上，这种几何结构几乎独一无二地决定了这一群体。一般而言，当一个群以几何形式作用于一个几何体时，通过跟随所有群元素下的一点的图像而创建的空间内有该群的‘’图像‘’。然后，最初是一个抽象的数学对象的群体，可以通过研究空间的几何来研究，其中的问题现在是几何问题，而不是代数问题。在上面的示例中，组由无限棋盘上从一个方块到另一个方块的移动组成。然后，通过识别每个正方形及其中心，动作的集合可以被赋予具体的几何图像，动作的整体变成离散地近似平面的晶格。***

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. ***