Asymptotic invariants of groups

研究人员研究群的渐近不变量。这些主题包括：构造具有“超越”性质(无限传递且具有Kazhdan性质(T))的有限表示群，双曲群的剩余性质和线性，具有不规则或小Dehn函数和群的其他填充函数的群，有限表示群的渐近圆锥，以及嵌入到Hilbert空间的渐近不变量(群的Hilbert空间压缩)。这些主题是密切相关的。例如，在Dehn函数的研究和双曲群的剩余性质的研究中，Higman嵌入和S机器被用来构造有限表示扭群。研究人员建议组织几次群论会议，吸引学生进入他们所在的数学领域。他们还将就提案的主题开设几门课程。在克莱因、希尔伯特、爱因斯坦和韦尔之后，基本定律描述了自然界中发生的对称性，这是一个众所周知的观点。对象的对称性可以通过与该对象对应的组来测量。群可以定义为对称的群，也可以抽象地通过算法描述(生成元和关系)来定义。在第二种方法中，研究人员选择一些基本对称(生成元)，使得所有其他对称都是基本对称的合成(词)，并描述基本对称之间的某些关系，使得所有其他关系都遵循所选择的关系。这种表示所给出的群的几何是用某些渐近不变量来描述的。自20世纪初M.Dehn的开创性工作以来，人们就已经知道了这些不变量，但研究人员发现了这些不变量与算法问题之间的深刻联系。研究人员正在发展他们的几何方法，解决算法性质的旧数学问题和相应的关于群的代数问题。

The investigators study asymptotic invariants of groups. The topics include:constructing finitely presented groups with "transcendental" properties (infinitetorsion and with Kazhdan property (T) ), residual properties and linearity ofhyperbolic groups, groups with irregular or small Dehn functions and other fillingfunctions of groups, asymptotic cones of finitely presented groups, and asymptoticinvariants of embeddings into Hilbert spaces (Hilbert space compressions of groups).These topics are intimately related. For example, Higman embeddings and S-machines are used to construct finitely presented torsion groups, in the study of Dehn functions,and in the study of residual properties of hyperbolic groups. The investigators proposeto organize several group theory conferences and attract students to their area ofmathematics. They are also going to give several courses on the topic of the proposal. It is a well known point of view after Klein, Hilbert, Einstein and Weil that fundamentallaws describe symmetries occuring in the nature. The symmetry of an object can be measured by the group corresponding to the object. Groups can be difined either as groups of symmetries or abstractly by an algorithmic description (generators andrelations). In the second approach, the investigators choose some basic symmetries(generators) so that all other symmetries are compositions (words) of the basic ones,and describe certain relations between the basic symmetries such that all otherrelations follow from the chosen ones. The geometry of groups given by suchpresentations is described in terms of certain asymptotic invariants. The invariantshave been known since the pioneering works of M.Dehn at the beginning of the 20thcentury, but the investigators discovered deep relationship between these invariantsand algorithmic problems. The investigators are developing their geometric methodsolving old mathematical problems of algorithmic nature and corresponding algebraicproblems about groups.