Cubical Geometry

数学是对模式的研究。最基本、最基本的图案类型是对称。有有限的对称“群”，如由立方体的48个不同对称组成的群，也有无限的对称群，如欧几里得平面的平方平铺的对称群。虽然对称的有限群可以很好地理解，但一些无限群是如此复杂，以至于它们显然是不可能理解的。然而，几何学和拓扑学中的许多重要问题都可以归结为理解某些无限几何对象的无限对称组。 这项研究旨在利用几何和拓扑推理来理解某些无限群。这项工作涉及到代数、几何和拓扑学之间的相互作用，并将阐明所有这三个学科。 所提出的研究的一个主要目的是考察非正曲线立方体复形在我们理解无限群和低维拓扑方面的应用。第二个目的是理解群的无处不在的‘局部拟凸’，即它们的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.