Boundaries of Nonpositively Curved Groups

小行星9704939 一类重要的群，几何思想已证明是有用的是一类字双曲群提出的M。 格罗莫夫 这些群离散地近似于一个更像双曲平面而不是欧几里得平面的几何。 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。 就像字双曲群是经典双曲群、n-生成自由群和某些小消去群的推广一样，也应该有一般的非正曲群，它包括n-生成自由交换群、更一般的小消去群和非正曲率黎曼流形的基本群。 最近，有几类非正曲群被提出。 其中一类由通过几何作用在“CAT（0）”空间上产生的群组成。 这些空间享有许多相同的几何性质的普遍覆盖的黎曼流形的非正曲率。 欧氏平面和双曲平面都是CAT（0）空间的例子。 CAT（0）空间中允许几何群作用的边界是这一领域的一个重要研究对象。 最近，群论中的几个重要问题已经用几何方法解决了。 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，在这些定理中，用“CAT（0）”代替“词双曲”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。 几何群论的基本思想是通过研究无限群G在不同几何上的“几何”作用来研究G的结构。 这样，任何这样的群都被看作是几何体的一组刚性运动。 要记住的一个例子是欧几里得平面。 这是一个几何体，其边界可以用单位圆来标识，其中圆的每个点代表平面中指向无穷远的方向。 直线上的刚体运动称为平移. 这个空间（平面）可以受到任何方向的平移的作用，但是两个独立方向的平移就足以给出所有的平移。 因此，整数群的两个（可交换的）副本可以被认为是通过平移作用在平面上。 事实上，这种几何结构几乎唯一地决定了这个群。 一般来说，当一个组以几何方式作用于一个几何体上时，在空间内有一个组的“图片”，该空间是通过跟随所有组元素下的一个点的图像而创建的。 然后，开始作为抽象数学对象的群可以通过研究空间的几何来研究，其中问题现在是几何的而不是代数的。 在上面的例子中，组由在无限棋盘上从一个正方形移动到另一个正方形组成。 然后，通过确定每个正方形的中心，移动的集合可以被赋予一个具体的几何图片，移动的整体成为离散近似平面的网格。 ***

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. ***