Topology and geometry of Cayley graphs for groups

群的凯莱图的拓扑和几何

• 批准号: 1313559
• 负责人: Susan Hermiller
• 金额: $ 24.6万
• 依托单位: University of Nebraska-Lincoln
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2013
• 资助国家: 美国
• 起止时间: 2013-08-15 至 2018-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1313559&HistoricalAwards=false
• 关键词: Topology; geometry; Cayley; graphs; groups

The objective of the proposed research is to investigate the interplay between topological/geometric properties of groups and algorithms for solving the word, conjugacy, and related problems. Major directions for this research include: (1) Applications of autostackable structures to 3-manifold groups, Thompson's group F, and metabelian groups; these structures are characterized using a combination of topological and formal language theory properties of the Cayley graph, and provide algorithms for solving the word problem and for building Dehn diagrams for the group. (2) A study of the asymptotic growth of conjugacy classes and of geodesic words and elements (up to conjugacy) of groups. (3) Improving understanding of filling invariants, by studying refinements of the isodiametric (i.e. intrinsic diameter) inequality and its extrinsic analog. (4) Applications of group invariants to problems in knot theory; local moves, in combination with the standard Reidemeister operations, will be studied to determine when these local moves are (or are not) unknotting or unlinking operations.The proposed work is centered around mathematical problems at the interface between group theory, geometry, and computer science. Group theory is the mathematical study of symmetries of objects; this research area has its beginnings in the study of crystal structures in chemistry, but now has wide applications across many disparate areas of mathematics and the sciences. Knot theory also has seen wide application in recent years, and in particular the study of unknotting via local moves is used to model enzyme actions on DNA molecules. The methods proposed make essential use of tools from one or more of these fields to shed light on important problems in another field. In particular geometric methods are used in the pursuit of computationally effective and efficient algorithms for groups, and algorithmic methods are used to determine geometric properties of groups, including applications to knot theory. This project includes mentoring of undergraduate and graduate students in mathematics research.

建议的研究的目的是调查的拓扑/几何性质的群体和算法之间的相互作用，解决字，共轭，和相关的问题。 本研究的主要方向包括：（1）自堆叠结构在3-流形群、Thompson群F和亚阿贝尔群中的应用;这些结构使用Cayley图的拓扑和形式语言理论性质的组合来表征，并提供解决字问题和为群构建Dehn图的算法。 (2)共轭类和测地线字与群元素（直到共轭）的渐近增长性的研究。(3)通过研究等径不等式（即内直径）及其外在类似物的改进，提高对填充不变量的理解。(4)群不变量在纽结理论问题中的应用;结合标准的Reidemeister操作，将研究局部移动，以确定这些局部移动何时是（或不是）解结或解链操作。拟议的工作围绕群论，几何和计算机科学之间的接口的数学问题。 群论是对物体对称性的数学研究;这个研究领域开始于化学中晶体结构的研究，但现在在数学和科学的许多不同领域都有广泛的应用。近年来，纽结理论也得到了广泛的应用，特别是通过局部移动解开纽结的研究被用来模拟酶对DNA分子的作用。所提出的方法基本上使用了这些领域中的一个或多个领域的工具，以阐明另一个领域中的重要问题。 特别是几何方法用于追求计算有效和高效的算法组，和算法的方法用于确定几何属性的群体，包括应用到纽结理论。 该项目包括指导本科生和研究生的数学研究。