Cubical Geometry

数学是对模式的研究。最基本和最基本的图案类型是对称。有有限的对称“群”，比如一个立方体的48种不同的对称组成的群，也有无限的对称群，比如欧几里得平面被正方形平铺的对称群。虽然有限群的对称性可以很好地理解，但一些无限群是如此复杂，以至于它们无法被证明是可以理解的。然而，几何和拓扑学中的许多重要问题可以归结为对某些无限几何对象的无限对称群的理解。本研究旨在利用几何和拓扑推理来理解某些无限群。这项工作涉及代数、几何和拓扑学之间的相互作用，并将阐明这三个学科。提出的研究的一个主要目的是检查非正弯曲立方体复合体在我们对无限群和低维拓扑的理解中的应用。第二个目标是理解“局部拟凸”群的普遍性，即它们所有有限生成的子群都是拟凸的。

Mathematics is the study of patterns. The most fundamental and elementary type of pattern is a symmetry. There are finite `groups' of symmetries, such as the group consisting of the 48 distinct symmetries of a cube, and there are also infinite groups of symmetries such as the group of symmetries of the tiling of the Euclidean plane by squares. While finite groups of symmetries can be understood quite well, some infinite groups are so complex that they are provably impossible to understand. Nevertheless, many important problems in geometry and topology can be reduced to understanding the infinite group of symmetries of some infinite geometric object. This research aims to understand certain infinite groups using geometric and topological reasoning. The work involves an interplay between algebra, geometry, and topology, and will shed light on all three disciplines. A primary aim of the proposed research is to examine applications of nonpositively curved cube complexes to our understanding of infinite groups and low-dimensional topology. A secondary aim is to understand the ubiquity of groups that are `locally quasiconvex' in the sense that all their finitely generated subgroups are quasiconvex.