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

9704417 Brady本项目的研究者将研究非正弯曲立方体配合物基本群的核子群的几何性质和有限性质。应用于立方体配合物的组合莫尔斯理论可以用来分析直角Artin群核的有限性质。其中一些核具有显著的有限性质。这个项目的一个主要目标是研究这些核的几何和拓扑结构。希望这些群包含区分几何维数和上同调维数的例子，以及区分可梳群和自动群的例子。同样值得考虑的是，莫尔斯理论论据可以应用在更一般的环境中，以及这些思想在研究各种类群的相干性问题时可能产生什么信息。另一个有趣的研究方向是莫尔斯技术是否可以与分支覆盖技术相结合，以产生不包含秩为2的自由阿贝尔子群的无扭群的例子，这些群的类型是FP(n)而不是FP（n+1）。本项目研究的主要对象是分段欧几里得立方复合体。它们可以被认为是由规则的欧几里得正方形和立方体构成的，沿着它们的面和边缘整齐地缝合在一起。这样的配合物具有对称群，可以被认为是墙纸图案群的概括，或者是人们在化学晶格研究中遇到的三维晶体群。本研究的主要目的是研究这些对称群的子群。一种技术是在立方体复合体上引入高度函数，并观察由所有这些对称组成的子群，这些对称保持了这些函数的“等高线”或水平集。这些等高的切片通常表现出非凡的几何图案，当人们仅仅观察周围的立方体复合体时，这些图案远不明显。保存这些切片的对称子群与欧几里得晶体学世界中遇到的对称子群具有非常不同的代数和几何性质。这些例子有助于加深我们对几何对称概念的理解。＊＊＊

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. ***