Mathematical Sciences: Geometric Group Theory and 3-Manifolds

群G关于一组生成元A的Cayley图是这样一个图，它的顶点是G的元素，使得对于G中的每个g和A中的每个生成元a，都有一条边从g到GA，用生成元a标记。这个图的拓扑和几何性质通常与群G的性质有关，而与生成元集无关。例如，格罗莫夫的“负曲线”群的概念可以用Cayley图来表述，即存在一个固定的数c，使得在Cayley图中的最短路径的每个三角形中，三角形的每条边都在其他两条边的并集的c-邻域内。还有各种其他性质可以用类似的术语来描述，尽管定义通常不是那么简单。其中最有趣的是斯蒂芬·G·布里克发明的一个性质，他称之为“QSF”；给出群表示，一个人构造了所描述的2-复形，将给定的群作为基本群；一个问题是，泛覆盖中单位的任意大邻域是否可以(在某种技术意义上)由映射到泛覆盖中的有限的1-连通复形来忠实地表示；如果是这样的话，这个群就被称为“准简单滤子”或QSF。推广了Poenaru和Casson的一些结果，从而得到一个定理：如果一个非球面的，P2-不可约的，闭的3-流形有QSF的基本群，那么它的泛覆盖是R3的同胚。在过去的几年里，这些概念和相关的概念在无限群理论中产生了一场革命。其基本模式似乎是在3-流形的微分几何和拓扑学中发现一些事实；人们抽象到基本群水平，然后发现群论定理，有时对拓扑学有相反的应用。此外，这个主题往往与Cayley图本身的可计算性有关；有限群的Todd-Coxeter算法因此扩展到关于有限Cayley图的各种可计算性问题。自动机理论涉及有限状态自动机和正则语言；然而，该理论不适用于某些容易理解的群，如整数上的幂零群和矩阵群；为了理解这些，人们需要以某些非常有限的方式扩展有限状态机的概念，这些方式还不明显。也许这会对形式语言理论和通常被认为是“计算机科学”的数学的其他方面产生影响。

The Cayley graph of a group G with respect to a set of generators A is the graph whose vertices are the elements of G, and such that for each g in G and each generator a in A, an edge runs from g to ga labeled with the generator a. This graph has topological and geometric properties which are often related to properties of the group G and independent of the set of generators. Gromov's notion of "negatively curved" group, for example, can be stated in terms of the Cayley graph as saying that there is a fixed number c such that in each triangle of shortest paths in the Cayley graph, each edge of the triangle is within the c-neighborhood of the union of the other two edges. There are various other properties which can be described in similar terms, although the definitions are not quite so simple, usually. One of the most interesting is a property invented by Stephen G. Brick which he calls "qsf"; given a group-presentation, one constructs the 2- complex which is described, having the given group as fundamental group; one asks whether arbitrarily large neighborhoods of the identity in the universal cover can be faithfully represented (in a certain technical sense) by finite, 1-connected complexes mapping into the universal cover; if so, the group is said to be "quasi- simply-filtrated" or qsf. There are generalizations of some results of Poenaru and Casson, so that one has a theorem to the effect that if an aspherical, P2-irreducible, closed 3-manifold has fundamental group which is qsf, then its universal cover is homeomorphic to R3. These and related notions have produced a revolution in the theory of infinite groups within the last few years. The basic pattern seems to be that one finds some facts in differential geometry and topology of 3-manifolds; one abstracts to the fundamental group level, and then one finds group-theoretic theorems which sometimes have a reverse application to topology. In addition, the subject tends to have a connection with the computability of the Cayley graph itself; the Todd-Coxeter algorithm for finite groups thus extends to various kinds of computability questions about finite Cayley graphs. The theory of "automatic groups" involves finite-state automata and regular languages; this theory, however, does not apply to certain easily understood groups such as nilpotent groups and matrix groups over the integers; in order to understand these, one needs to extend the notion of a finite-state machine in certain very restricted ways which are not yet obvious. Perhaps this will have implications in formal language theory and other aspects of mathematics usually considered to be "computer science."