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

9704417 Brady，这个项目的研究者，将研究非正曲线立方体复形的基本群的核子群的几何性质和有限性质。应用于立方体复形的Morse理论的组合形式可以用来分析直角Artin群的核的有限性质。其中一些核函数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内核的几何和拓扑。希望这些群包含区分几何和上同调维度的例子，以及区分可梳群和自动群的例子。同样值得考虑的是，可以应用莫尔斯理论论证的更一般的环境，以及这些想法在研究各类群体的连贯性问题时可能产生的信息。另一个有趣的研究方向是Morse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的自由交换子群的扭自由群的例子，并且是类型为FP(N)但不是FP(N 1)的群。本项目研究的主要对象是分段欧几里德立方复形。这些可以被认为是由规则的欧几里得正方形和立方体组成的，沿着它们的面和边整齐地缝合在一起。这种络合物具有对称性群，这可以被认为是墙纸图案群的推广，或者是人们在化学中研究晶格时遇到的三维结晶学群。本研究的主要目的是考察这些对称群的子群。一种技术是在立方复形上引入高度函数，并查看由所有这些对称性组成的子群，这些对称性保持这种函数的‘轮廓线’或水平集。这些固定高度的切片经常显示出引人注目的几何图案，当你只看周围的立方体建筑群时，这些图案是远远不明显的。保持这些薄片的对称子群的代数和几何性质与人们在欧几里得结晶学世界中遇到的完全不同。这些例子有助于加深我们对几何对称性概念的理解。***

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. ***