Asymptotic and algorithmic properties of groups

群的渐近性质和算法性质

• 批准号: 0245600
• 负责人: Mark Sapir
• 金额: $ 27.15万
• 依托单位: Vanderbilt University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2003
• 资助国家: 美国
• 起止时间: 2003-07-01 至 2006-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0245600&HistoricalAwards=false
• 关键词: Asymptotic; algorithmic; properties; groups

DMS-0245600Sapir, Mark V.AbstractTitle: Asymptotic and algorithmic properties of groupsThe PIs propose to study asymptotic and algorithmic properties of groups.The topics include a torsion free non-amenable finitely presented groupwithout free non-abelian subgroups, a finitely presented infinite torsion group, Dehn functions of groups including Dehn functions of residually finite and Metabelian groups, isoperimetric functions of aspherical manifolds, residually finite hyperbolic groups, structure properties of hyperbolic groups (is everytorsion-free hyperbolic group free-by-finite exponent?), n-dimensionaldiagram groups.Algorithmic properties of groups have been a subject of intensive studysince the beginning of the 20th century (Dehn and Tietze). Novikov, Booneand Higman showed deep connections between logic (especially the theory ofrecursive functions) and group theory. A more recent work by Gromov andothers showed a connection between algorithmic and asymptotic properties(especially isoperimetric and growth functions) of groups and the relatedtopological objects. The PIs found even more intimate connections betweencomplexity of algorithms and asymptotic properties of groups. In particular,they have characterized groups whose word problem is in NP in terms of Dehnfunctions and Higman embeddings, found an NP-complete group, etc.. The PIspropose to further study these connections, and their applications to someoutstanding Burnside-type problems. In particular, they propose to useHigman embeddings to construct finitely presented torsion groups and in thestudy of Dehn functions of metabelian groups. Another direction of theirresearch deals with connections between geometry and structure of groups. Inparticular, they propose to study the class of finite dimensional diagramgroups, for dimension greater than two which, they believe, include a large class of groups acting ``nicely" on cubical complexes. Structural properties of hyperbolic groups will also be under investigation.

sapir, Mark v .[摘要]题目：群的渐近和算法性质PIs提出研究群的渐近和算法性质。题目包括无自由非阿贝尔子群的无扭转非可服从有限呈现群、无有限呈现无限扭转群、群的Dehn函数（包括剩余有限群和亚贝群的Dehn函数）、非球面流形的等周函数、剩余有限双曲群、双曲群的结构性质（是否所有无扭转双曲群都是自由的？）、n维图群。自20世纪初以来，群的算法特性一直是一个深入研究的主题（Dehn和Tietze）。诺维科夫、布恩和希格曼展示了逻辑（尤其是递归函数理论）和群论之间的深刻联系。Gromov和其他人最近的工作表明了群和相关拓扑对象的算法和渐近性质（特别是等周函数和生长函数）之间的联系。pi发现算法的复杂性和群的渐近性质之间有更密切的联系。特别地，他们用Dehnfunctions和Higman embedding刻画了NP中的词问题群，发现了NP完全群等。研究人员建议进一步研究这些联系，并将其应用于一些突出的burnside型问题。特别是，他们提出使用higman嵌入来构造有限表示的扭转群，并在亚群的Dehn函数的研究中。他们研究的另一个方向是几何和群结构之间的联系。特别是，他们提议研究一类有限维图群，对于大于2维的图群，他们认为，这类图群包含了一大类在立方复合体上“很好地”作用的群。双曲群的结构性质也将被研究。