Mathematical Sciences: Geometric Group Theory and 3-Manifolds

小行星9503034 该项目是研究双曲群，自动群，群的非正曲三角形，以及涉及群论几何问题的相关事项。 区分非双曲自动群的问题有几个有趣的方面。 有一个概念之间的角度对子群A，B的一个组G，其中涉及的最小长度的表达式aba 'B'a“B”.它被约化并给出G的单位元。 这导致了有趣的算法问题;在自由群的情况下，这涉及到循环定理。 另一个关于“内秩”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，即保持结构的变换。 研究者希望了解群体的各种性质，群体类型的分类，以及使用群论来连接数学领域。 在几何学和拓扑学中，群自然地作为矩阵群和空间的基本群的子群出现。 庞加莱定义的基本群是三维流形分类的基本组成部分，主要问题是群论。 在三维空间中尤其如此。 在抽象群中也有一个内在的大尺度几何，这导致了最近对所谓的双曲群的兴趣。 这反过来又导致算法问题和形式语言理论，并将有一天感兴趣和使用的计算机科学家。 该项目已经确定了一些问题领域如上所述，“角度”，“内部排名”，等等，这是感兴趣的互动代数和几何;在这些领域已经取得了一些进展，他们正在成为活跃的研究领域，特别是由调查员和他的博士。 学生 ***

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. ***