Mathematical Sciences: The Geometry of Kernel Subgroups of Nonpositively Curved Cube Complex Groups

数学科学：非正曲立方复群核子群的几何

• 批准号: 9704417
• 负责人: Noel Brady
• 金额: --
• 依托单位: Cornell 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 1999-07-23
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9704417&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Geometry; Kernel; Subgroups

9704417 Brady The investigator of this project will study the geometric and finiteness properties of kernel subgroups of the fundamental groups of nonpositively curved cube complexes. A combinatorial version of Morse theory applied to cube complexes may be used to analyze the finiteness properties of the kernels of right-angled Artin groups. Some of these kernels have remarkable finiteness properties. A major goal of this project is to study the geometry and topology of these kernels. It is hoped that these groups contain examples which distinguish between geometric and cohomological dimension, and examples which distinguish between combable and automatic groups. It is also worth considering more general settings in which the Morse theory arguments may be applied, and what information these ideas may yield in the study of the coherency question for various classes of groups. Another interesting direction to investigate is whether the Morse techniques may be combined with branched covering techniques to produce examples of torsion-free groups that do not contain free abelian subgroups of rank 2, and that are of type FP(n) but not FP(n+1). The main objects under investigation in this project are piecewise euclidean cubical complexes. These may be thought of as constructed from regular euclidean squares and cubes, neatly stitched together along their faces and edges. Such complexes have symmetry groups, which may be thought of as generalizations of the wallpaper pattern groups, or of the 3-dimensional crystallographic groups one encounters in the study of lattices in chemistry. The main purpose of this research is to examine subgroups of these symmetry groups. One technique is to introduce height functions on the cubical complexes and to look at the subgroups consisting of all those symmetries that preserve the `contour lines' or level sets of such functions. These constant-height slices often exhibit remarkable geometric patterns that are far f rom obvious when one looks just at the ambient cubical complex. The subgroups of symmetries that preserve these slices have very different algebraic and geometric properties from those one encounters in the euclidean crystallographic world. These examples help deepen our understanding of the concept of geometric symmetry. ***

9704417布雷迪 本计画研究非正曲立方复形基本群的核子群的几何性质与有限性。 应用于立方复形的组合形式的莫尔斯理论可以用来分析直角Artin群的核的有限性。 其中一些内核具有显着的有限性属性。 该项目的一个主要目标是研究这些内核的几何形状和拓扑结构。 人们希望这些群体包含的例子区分几何和上同调尺寸，并区分combable和自动组的例子。 还值得考虑的是莫尔斯理论论证可能适用的更一般的设置，以及这些思想在研究各类群的相干性问题时可能产生的信息。 另一个有趣的研究方向是莫尔斯技巧是否可以与分支覆盖技巧相结合，以产生不包含秩为2的自由阿贝尔子群的无挠群的例子，并且是FP（n）型但不是FP（n+1）型。 该项目研究的主要对象是分片欧几里得三次复形。 这些可以被认为是 由规则的欧几里得正方形和立方体构成，沿着它们的面和边沿着整齐地缝合在一起。 这种配合物具有对称群，可以认为是墙纸图案群的推广，或者是在化学晶格研究中遇到的三维晶体群的推广。 本研究的主要目的是研究这些对称群的子群。 一种技术是引入高度函数的立方复合体，并期待在所有这些对称性，保持'轮廓线'或水平集的功能组成的子群。 这些 恒定高度的切片经常表现出显著的几何图案，当人们只看周围的立方体复合体时，这些几何图案远不明显。 保留这些切片的对称子群具有与那些切片非常不同的代数和几何性质 在欧几里得晶体学世界中的遭遇。 这些例子有助于加深我们对几何对称概念的理解。 ***