Large-scale geometry of groups and spaces with nonpositive curvature

建议：DMS-9972047PI：Bruce Kleine摘要：建议的研究是在群论、微分几何、动力学和拓扑学之间的相互作用的领域，这些领域是在过去100年里从研究负曲率曲面发展而来的。通过考虑离散群--例如紧致流形的基本群--根据它们的Cayley图，人们可以将它们视为本身的几何对象。贯穿整个提案的这一观点导致了一些引人注目的见解(Mostow刚性、Stallings定理、关于多项式增长群的Gromov定理等)。令人惊讶的事实是，许多离散的群(例如单词双曲群)以规范的方式在“边界”上起作用。拟研究的第一部分集中在双Lipschitz映射上。该项目的一个副产品，如果它是成功的，将是对以下问题的答案：何时平面的某些非周期性瓷砖(例如，PenRose瓷砖)在弱组合上等价于通常的正方形瓷砖，即人们可以匹配两个瓷砖的瓷砖，从而使相邻瓷砖与附近的瓷砖匹配。第二个项目解决了Solv，这是唯一一个从大尺度角度看仍然神秘的瑟斯顿三维几何图形。剩下的主题涉及非正弯曲空间(测地线流的动力学和基本群的结构)、双曲群(边界的拓扑及其与群结构的关系)和Poincare对偶群(Haken PD(3)群和3-流形群之间的关系)等领域的重要问题。该提案有两个主要主题：对称性、分类和瓷砖图案的识别；以及动力学。平面和三维空间的拼贴已经被深入研究了几个世纪，由此产生的理论是许多数学分支的基础，也是固体物理(晶体结构)的基础。在20世纪70年代的S中，数学家们发现了瓷砖的惊人的新应用(特别是双曲空间的瓷砖，这是埃舍尔许多图像背后的几何图形)，通过使用瓷砖的“粗略”图案，而不是瓷砖及其相交图案的精确形状。从这个新的观点来看，如果一个瓦片的瓦片与另一个瓦片的瓦片之间存在一对一的对应关系，使得一个瓦片中的相邻瓦片与另一个瓦片中的邻近瓦片相对应，则两个瓦片将被视为“大致相同”。这是一个非常活跃的研究领域，与许多数学领域有联系。一些拟议的研究主题与此相关：例如，如果与准晶体结构相关的晶格总是与经典晶体晶格大致相同，这是一个悬而未决的问题。提案中的另一个主题是测地线运动--自由粒子在特定类型的空间中的运动。这些系统有很长的历史，提供了特别简单的“混沌”机械系统的例子；它们还可以用来模拟一个由几个台球组成的系统，该系统在带有圆形(或凸起)保险杠的台球桌上。这里的目标是至少在某些类别的例子中描述和分类系统的所有可能的轨迹。

Proposal: DMS-9972047PI: Bruce KleinerAbstract: The proposed research is in the area of interaction between group theory, differential geometry, dynamics, and topology which evolved during the last 100 years from the study of surfaces with negative curvature. By thinking of discrete groups -- such as fundamental groups of compact manifolds -- in terms of their Cayley graphs, one can view them as geometric objects in their own right. This viewpoint, which is present throughout the proposal, leads to some remarkable insights (Mostow rigidity, Stallings' theorem, Gromov's theorem on groups of polynomial growth, etc.) and to the striking fact that many discrete groups (for example word hyperbolic groups) act in a canonical way on a ``boundary''. The first part of the proposed research focusses on biLipschitz maps. A byproduct of the project, if it is successful, will be an answer to the question of when certain aperiodic tilings of the plane (for example, Penrose tilings) are weakly combinatorially equivalent to the usual square tiling, in the sense that one can match the tiles of the two tilings so that adjacent tiles are matched with nearby tiles. The second project addresses Solv, the only 3-dimensional Thurston geometry that remains mysterious from a large-scale viewpoint. The remaining topics concern important issues in the fields of nonpositively curved spaces (the dynamics of the geodesic flow and the structure of the fundamental group), hyperbolic groups (the topology of the boundary and its relation to group structure), and Poincare duality groups (the relation between Haken PD(3) groups and 3-manifold groups). The proposal has two main themes: symmetry, classification, and recognition of tiling patterns; and dynamics. Tilings of the plane and 3-space have been studied intensely for centuries, and the resulting theory is fundamental in many branches of mathematics, as well as in solid state physics (crystal structure). In the 1970's, mathematicians discovered striking new applications of tilings (especially tilings of hyperbolic space, the geometry underlying a number of Escher's images), by using the ``rough'' pattern of the tilings rather than the precise shape of the tiles and their intersection pattern. From this new viewpoint two tilings would be considered "roughly the same" if there is a one-to-one correspondence between the tiles of one tiling with the tiles of the other so that adjacent tiles in one tiling correspond to nearby tiles in the other. This is a very active area of research with connections to many areas of mathematics. Several of the proposed research topics relate to this: for example it is an open problem if the tilings associated with quasi-crystal structure are always roughly the same as classical crystal tilings. Another theme in the proposal is geodesic motion -- the motion of a free particle -- in a certain class of spaces. These systems have a long history and provide especially simple examples of ``chaotic'' mechanical systems; they can also be used to model a system of several billiard balls on a billiard table with circular (or convex) bumpers. The objective here is to describe and classify, at least in certain classes of examples, all possible trajectories of the system.