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

点击翻译按钮获取中文摘要

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. ***