Asymptotic and algorithmic properties of groups

DMS-0245600 Sapir，Mark V.摘要标题：群的渐近性质和算法性质PI建议研究群的渐近性质和算法性质。主题包括没有自由非交换子群的无挠非顺从有限表示群，有限表示无限挠群，群的Dehn函数，包括剩余有限群和亚阿贝尔群的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(每个扭转自由的双曲群是按有限指数自由的吗？)，n维图群。自20世纪初以来，群的算法性质一直是一个密集的研究主题(Dehn和Tietze)。诺维科夫、布恩和希格曼展示了逻辑(特别是递归函数论)与群论之间的深刻联系。Gromov等人最近的一项工作表明，群和相关拓扑对象的算法性质和渐近性质(特别是等周函数和增长函数)之间存在联系。PI发现算法的复杂性和群的渐近性质之间甚至有更密切的联系。特别地，他们利用DehnFunction和Higman嵌入刻画了字问题在NP中的群，找到了NP-完全群等。PIS建议进一步研究这些联系，以及它们在一些悬而未决的Burnside类型问题中的应用。特别地，他们提出利用Higman嵌入来构造有限表示扭群，并用于亚交换群的Dehn函数的研究。他们研究的另一个方向是群的几何和结构之间的联系。特别是，他们建议研究维度大于2的有限维图群，他们认为，这包括一大类作用在立方复形上的群。双曲群的结构性质也将被研究。

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.