Asymptotic and algorithmic methods in group theory

群论中的渐近方法和算法方法

• 批准号: 1161294
• 负责人: Mark Sapir
• 金额: $ 43.28万
• 依托单位: Vanderbilt University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2012
• 资助国家: 美国
• 起止时间: 2012-06-01 至 2016-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1161294&HistoricalAwards=false
• 关键词: Asymptotic; algorithmic; methods; group; theory

The proposal describes several projects related to algorithmic and asymptotic aspects of group theory: * Constructing finitely presented groups with "transcendental" properties (in particular, finitely presented infinite torsion groups). * Complexity of residually finite groups. * Asymptotic and Burnside properties of group actions of maximal growth * Asymptotic cones of finitely presented groups. * Asymptotic properties of amenable groups.These topics are intimately related. For example, Higman embeddings and S-machines are going to be used to construct finitely presented torsion groups and in the study of complexity of the word problem in groups.Modern group theory is a very fast developing area of mathematics that accumulates methods from geometry, combinatorics, logic and computer science. Our proposal describes projects that are require methods and ideas from all these areas. In particular, constructing a finitely presented infinite torsion group (which is one of the outstanding problems in group theory) will require constructing a suitable kind of Turing machine (called S-machines) and adapting methods from geometric group theory to find suitable quotients of groups simulating these machines. Another idea coming from geometry (and due mostly to Gromov) is the idea of an asymptotic cone of a group. We are going to study asymptotic cones to discover algebraic and algorithmic properties of groups.

该提案描述了与群论的算法和渐近方面相关的几个项目： * 构造具有“超越”属性的有限呈现群（特别是有限呈现的无限挠场群）。 * 剩余有限群的复杂性。 * 最大增长群作用的渐近和伯恩赛德性质 * 有限呈现群的渐近锥。 * 顺应群的渐近性质。这些主题密切相关。例如，希格曼嵌入和 S 机将用于构造有限呈现的挠群以及研究群中问题的复杂性。现代群论是一个快速发展的数学领域，积累了几何学、组合学、逻辑学和计算机科学的方法。我们的提案描述了需要所有这些领域的方法和想法的项目。特别是，构造一个有限呈现的无限挠场群（这是群论中的突出问题之一）将需要构造一种合适的图灵机（称为 S 机）并采用几何群论的方法来找到模拟这些机器的合适群的商。来自几何学的另一个想法（主要归功于格罗莫夫）是群的渐近锥体的想法。我们将研究渐近锥以发现群的代数和算法性质。