Mathematical Sciences: Geometric Group Theory and 3-Manifolds

9503034停顿这个项目是研究双曲群、自动机群、群的非正曲线三角形以及涉及群论几何问题的相关问题。区分非双曲自动组的问题有几个有趣的方面。存在群G的子群A、B对之间的夹角的概念，它涉及表达式Aba‘b’a“b”的最小长度…它被简化，并给出了G的恒等式元素。这导致了有趣的算法问题；在自由群的情况下，这与环路定理有关。另一个关于“内阶”的算法问题与三维流形中的曲面问题有关。群出现在数学的所有分支中，作为对称的集合，即保持结构不变的变换。研究人员希望了解群的各种性质，群类型的分类，以及使用群论将数学领域联系起来。在几何和拓扑学中，群自然而然地作为矩阵群和基本空间群的子群出现。Poincare定义的基本群是三维流形分类的基本成分，主要问题是群论。这在第三维空间中尤其如此。在抽象群中还有一种内在的大尺度几何，这导致了最近人们对所谓的双曲群的兴趣。这反过来又导致了算法问题和形式语言理论，总有一天会引起计算机科学家的兴趣和使用。该项目已经确定了一些对代数和几何的相互作用感兴趣的问题领域，如“角度”、“内部等级”等；在这些领域已经取得了一些进展；它们正在成为研究人员和他的博士生的活跃领域。***

9503034 Stallings The project is to study hyperbolic groups, automatic groups, non-positively curved triangles of groups, and related matters involving geometric questions about group-theory. The problem of distinguishing non-hyperbolic automatic groups has several interesting aspects. There is a notion of angle between pairs of subgroups A,B of a group G, which involves the smallest length of an expression aba'b'a"b"... which is reduced and gives the identity element of G. This leads to interesting algorithmic questions; in the case of free groups this relates to the Loop Theorem. Another algorithmic question about "inner rank" is related to questions about surfaces in 3-manifolds. Groups arise in all branches of mathematics, as sets of symmetries, i.e. transformations which preserve structure. The investigator hopes to understand various properties of groups, the classification of types of groups, and the use of group theory to connect mathematical areas. In geometry and topology, groups naturally arise as subgroups of matrix groups and the fundamental groups of spaces. The fundamental group as defined by Poincare is the basic ingredient in the classification of three-dimensional manifolds, and major problems are group-theoretic. This is especially true in dimension three. There is also an intrinsic large-scale geometry inherent in an abstract group, which has led to the recent interest in what are known as hyperbolic groups. This leads in turn to algorithmic problems and to formal language theory and will some day be of interest and use to computer scientists. The project has identified some problem areas as described above, "angle," "inner rank," and so on, which are of interest in the interaction of algebra and geometry; some progress in these areas has been made; they are becoming active areas of research, in particular, by the investigator and his Ph.D. students. ***