Cubical Geometry

立方几何

• 批准号: 238946-2013
• 负责人: Wise, Daniel
• 金额: $ 2.77万
• 依托单位: McGill University
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2017
• 资助国家: 加拿大
• 起止时间: 2017-01-01 至 2018-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=635195
• 关键词: Cubical; Geometry

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.

数学是对模式的研究。最基本和基本类型的模式是对称性。有有限的对称性组，例如由多个立方体的48个不同的对称性组成的组，并且还有一个无限的对称性组，例如通过正方形的欧几里得平面平铺的对称性组。尽管有限的对称组可以很好地理解，但一些无限的群体是如此复杂，以至于不可能理解它们。然而，可以简化几何和拓扑中的许多重要问题，以理解某些无限几何对象的无限对称性群体。这项研究旨在使用几何和拓扑推理来理解某些无限群体。这项工作涉及代数，几何和拓扑之间的相互作用，并将阐明这三个学科。拟议的研究的主要目的是检查非弯曲的立方体复合物的应用在我们对无限基团和低维拓扑的理解中。第二个目的是了解其所有有限生成的亚组都是Quasiconvex的意义上，这些群体的普遍存在。