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

9704417 Brady 该项目的研究者将研究非正弯曲立方体复合体基本群的核子群的几何和有限性性质。 应用于立方体复合体的莫尔斯理论的组合版本可用于分析直角Artin群的核的有限性性质。 其中一些内核具有显着的有限性特性。 该项目的主要目标是研究这些内核的几何形状和拓扑结构。 希望这些群包含区分几何维数和上同调维数的示例，以及区分可梳理群和自动群的示例。 还值得考虑可以应用莫尔斯理论论证的更一般的设置，以及这些想法在研究不同类别群体的一致性问题时可以产生哪些信息。 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来产生不包含 2 阶自由阿贝尔子群且类型为 FP(n) 但不是 FP(n 1) 的无扭转群的示例。 该项目研究的主要对象是分段欧几里得立方复形。 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，沿着它们的面和边缘整齐地缝合在一起。 这种配合物具有对称群，这可以被认为是壁纸图案群的概括，或者是化学晶格研究中遇到的 3 维晶体学群的概括。 这项研究的主要目的是检查这些对称群的子群。 一种技术是在立方复形上引入高度函数，并查看由保留此类函数的“等高线”或水平集的所有对称性组成的子群。 这些恒定高度的切片通常表现出显着的几何图案，当人们仅观察周围的立方体复合体时，这些图案远非显而易见。 保留这些切片的对称子群具有与欧几里得晶体学世界中遇到的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. ***