Asymptotic invariants of groups

研究者研究群的渐近不变量。主题包括：构造具有“超越”性质的有限呈现群（无限挠性和Kazhdan性质(T)），双曲群的残差性质和线性，具有不规则或小Dehn函数的群和群的其他填充函数，有限呈现群的渐近锥，嵌入Hilbert空间的渐近不变量（群的Hilbert空间压缩）。这些话题是密切相关的。例如，Higman嵌入和s机用于构造有限呈现的扭转群，用于研究Dehn函数，以及用于研究双曲群的剩余性质。研究人员建议组织几次群论会议，吸引学生到他们的数学领域。他们还将就提案的主题开设几门课程。这是继克莱因、希尔伯特、爱因斯坦和韦尔之后的一个著名观点，即基本定律描述了自然界中发生的对称性。一个物体的对称性可以通过该物体对应的群来测量。群既可以定义为对称群，也可以通过算法描述（生成器和关系）抽象地定义。在第二种方法中，研究者选择一些基本对称（生成器），以便所有其他对称都是基本对称的组成（词），并描述基本对称之间的某些关系，以便所有其他关系都遵循所选择的对称。用一定的渐近不变量来描述由这种表示给出的群的几何。自20世纪初dehn先生的开创性工作以来，不变量已经为人所知，但研究人员发现了这些不变量与算法问题之间的深刻关系。研究人员正在发展他们的几何方法，解决算法性质的老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.