Asymptotic invariants of groups

研究者研究群的渐近不变量。内容包括：构造具有超越性质（无穷挠和Kazhdan性质（T））的群，双曲群的剩余性质和线性性，具有非正则或小Dehn函数和其它填充函数的群，群的渐近锥，嵌入Hilbert空间的渐近不变量（群的Hilbert空间压缩），这些内容是密切相关的。例如，Higman嵌入和S-机被用于构造双曲表示扭群、Dehn函数的研究和双曲群的剩余性质的研究。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。他们还将就该提案的主题开设几门课程。基本定律描述了自然界中存在的对称性，这是继Klein、Hilbert、Einstein和Weil之后的一个著名观点。一个物体的对称性可以通过对应于该物体的群来测量。群既可以定义为对称群，也可以通过算法描述（生成元和关系）进行抽象。在第二种方法中，研究者选择一些基本对称性（生成元），使所有其他对称性都是基本对称性的组合（词），并描述基本对称性之间的某些关系，使所有其他关系都遵循所选的对称性。由这种表示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，不变量就已为人所知，但研究人员发现这些不变量与算法问题之间有着深刻的联系。研究人员正在发展他们的几何方法，解决古老的数学问题的算法性质和相应的代数问题的群体.

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.