NSF/CBMS Regional Conference in the Mathematical Sciences -Non-Positive Curvature in Group Theory; Albany, NY; August 15-20, 2004

我们开始讨论的起源几何群论：重要性的核心决策问题首先制定的德恩;出现类的群体，需要特别注意;和出现的负面和非积极的曲率从核心问题在群论。这导致了一个描述的地理的宇宙的非正曲率的表现形式所发挥的作用，强调的描述。从这个平台上，以下主题将得到发展。研究和建设的群体行动;复杂的群体，非正曲率的作用indevlopability。非正曲（Orbi）空间的基本群。刚度维数的概念。双曲群。朝向半双曲群：竞争理论与区分群的相关类的困难。丰富多样的子群理论。数学结构：通过图表将基本决策问题的复杂性与流形几何联系起来; Dehn函数和等周谱;语言理论的复杂性，规范形式的几何，自动群，共轭问题和密码学，非正曲率存在下群和流形的同构问题（积极和消极的结果）。在整个过程中，许多例子将被给予补充优雅的一般理论，并说明如何从思想的非正曲率在群论可以适用于其他地方，如解决格罗滕迪克的问题profinite完善。

We begin with a discussion of the origins of geometric group theory: the importance of the core decision problems first formulated by Dehn; the emergence of classes of groups that demand special attention; and the emergence of negative and non-positive curvature from the core issues in group theory. This leads to a description of the geography of the universe of finitely presented groups, with emphasis on the role played by manifestations of non-positive curvature.From this platform, the following topics will be developed.The study and construction of group actions; complexes of groups, the role of non-positive curvature indevlopability. The fundamental groups of non-positively curved (orbi)spaces. Rigidity. Concepts of dimension.Hyperbolic groups. Towards semihyperbolic groups: competing theories and the difficulties of discriminating between the associated classes of groups. The rich and diverse subgroup theory.Algorithmic structures: connecting the complexity of the basic decision problems to the geometry of manifolds via diagrams; Dehn functions and the isoperimetric spectrum; language-theoretic complexity, the geometry of normal forms, automatic groups.Conjugacy problems and cryptography.Isomorphism problems for groups and manifolds in the presence of non-positive curvature (positive and negative results).Throughout, many examples will be given to complement the elegance of the general theory and to illustrate how ideas from non-positive curvature in group theory can be applied elsewhere, eg the solution to Grothendieck's problems concerning profinite completions.