Asymptotic and Algorithmic Invariants of Groups

研究了群的渐近性质和算法性质。研究领域包括群的Dehn函数和相对Dehn函数，群的算法问题的复杂性，群的Burnside问题。近四十年来发表的许多论文都揭示了群的渐近不变量与拓扑、几何、适应性、动力系统等学科之间的深刻联系。最近，作者（与Rips和Birget一起）发现了等周函数的渐近性与群的词问题的复杂性之间的密切关系。研究人员正在开发他们的方法，结合几何和计算方法。他们的目标之一是利用几何群论获得其他算法问题的解。特别是研究者相信他们将能够证明每一个递归表示的有限生成群都可以嵌入到一个有限表示群中，并且具有相同程度的共轭问题的不可解性。这将解决柯林斯在1976年提出的一个问题。研究人员的计算几何方法也显示出能够有效地应用于burnside型问题的希望。研究人员接近于构造没有非循环自由子群的非可服从有限呈现群。这将解决冯·诺伊曼1929年论文中相应的著名问题。由于所讨论的群是有限指数的涡旋群通过循环群的有限呈现扩展，因此这个例子将是寻找有限呈现扭转群（无限群理论中的主要开放问题之一）的一个突破。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）抽象地定义为群。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关系。关于由生成器、关系和关系给出的群的一个主要问题是字问题：何时两个生成器的组合相同？在某些特殊情况下，这个问题可能是无法确定的，也就是说，对于某些组，没有自动过程来识别生成器的两个单词是否相等。但是，即使在这个问题可以确定的情况下，自动程序也可能非常复杂。近年来，研究者们发现群的字问题与群的整体几何之间有很深的关系。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，不变量已经为人所知，但研究者发现了这些不变量与算法问题之间的深刻关系。研究者正在发展他们的几何方法来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 and algorithmic properties of groups. The area of research includes Dehn functions and relative Dehn functions of groups, the complexity of algorithmic problems of groups, Burnside problems for groups. Many papers published during the last forty years showed deep connection between asymptotic invariants of groups and topology, geometry, amenability, dynamical systems, etc. Quiterecently, the proposers (jointly with Rips and Birget) discovered a closerelationship between the asymptotics of isoperimetric functions, on the onehand, and the complexity of the word problem for groups, on the other hand.The investigators are developing their method, combining geometric andcomputational approaches. One of their goals is to obtain the solutionof other algorithmic problems using geometric group theory. In particularthe investigators believe they will be able to prove that every recursively presentedfinitely generated group can be embedded into a finitely presented groupwith the same degree of unsolvability of the conjugacy problem. This wouldsolve a problem formulated by Collins in 1976.The computational-geometric methods of the investigators also show promise of being able to be fruitfully applied to the Burnside-type problems.The investigators are close to constructing non-amenable finitely presentedgroups having no non-cyclic free subgroups. This will solve thecorresponding well-known problem which goes back to von Neumann's 1929paper. Since the group in question is a finitely presented extension of atorsion group of finite exponent by a cyclic group, this example will bea breakthrough in the search for a finitely presented torsiongroup (one of the main open problem in the theory of infinite groups). It is a well known point of view after Klein, Hilbert, Einstein and Weilthat fundamental laws describe symmetries occurring in the nature. Thesymmetry of an object can be measured by the group corresponding to theobject. Groups can be defined as groups of symmetries or abstractly by analgorithmic description (generators and relations). In the second approach,the investigators choose some basic symmetries (generators) so that all other symmetriesare compositions (words) of the basic symmetries, and describe certainrelations between the basic symmetries such that all other relations followfrom the chosen relations. One of the main problems about a group given bygenerators and relations and relations is the word problem: when are twocompositions of generators the same? In some exotic cases this problem canbe undecidable, that is there are groups for which there are no automaticprocedures to recognize if two words of generators are equal. but evenin cases when this problem is decidable, the automatic procedure can be verycomplicated. In recent years the investigators discovered a deeprelationship between the word problem of a group and the global geometry ofthe group. The geometry of a group is described in terms of certainasymptotic invariants. The invariants have been known since the pioneeringworks of M. Dehn at the beginning of the 20th century but the investigatorsdiscovered deep relationship between these invariants and algorithmicproblems. The investigators are developing their geometric method solvingold mathematical problems of algorithmic nature and corresponding algebraic problems about groups.