Mathematical Sciences: Geometric Topology and the Fundamental Group

大多数（也许所有）三维流形都是几何的。 几何群 理论试图将组合性质和几何性质联系在一起 几何等距群。 这一理论正在蓬勃发展， 格罗莫夫，爱泼斯坦，瑟斯顿，霍尔特，贝斯特，费恩，梅斯， 帕特森，格斯滕，肖特，夏皮罗，帕里，弗洛伊德，鲍姆斯拉格，卡森， 布里克，斯塔林斯，整个欧洲数学家学校， 大炮和其他 该理论产生了以下概念： 自动组、负曲组、复合组、几乎 凸群，可梳群，等周和等径 群体的不平等。 在这些众多的话题中，坎农，他的 学生和同事目前正在集中精力研究一组 项目有关的双曲结构的存在3- 流形 一般的双呈现组是消极的吗 曲，声称没有明确的证据格罗莫夫？ 是不是每个 负曲闭3-流形允许一个黎曼度量， 常负曲率 负曲率能被识别吗 在算法上？ 无限远的空间能被认识吗 在算法上？ 潜常曲率能被识别吗 在算法上？ 坎农正在使用这些研究的基础 他的组合黎曼映射定理发展于 以前的项目，帕里的相关工作，以及计算机 爱泼斯坦和他的合作者的计划。 群是描述 精确对称的概念。 为此， 在几何学（和拓扑学）中的广泛作用。 虽然更常见的是 群论和几何学之间的相互作用是使用 代数计算涉及群体证明定理 几何物体具有对称性，有一个非常重要的 数学分支学科中的角色是颠倒的，我们的 对几何物体的直觉帮助我们理解和 证明关于相联群的定理。 最近一个令人着迷的 发展是计算机在这一切中的影响。 不仅 它是否是每个人都知道的无处不在的时间节省器，使人们能够 进行计算，否则将过于艰巨，但 它实际上影响了几何群论中的问题 在更基本的方面。 某些类别的群体 根据假设的机器类型来定义， 能够对它们进行某些计算。 真实的硬件 并没有涉及到这一点，同样的定义可能已经被 很久以前就有了，但他们不是，因为存在， 计算机已经影响了我们看待世界的方式， 我们问的问题的类型，我们问的性质的类型， 发现有趣。

Most (perhaps all) 3-manifolds are geometric. Geometric group theory seeks to tie together combinatorial and geometric properties of geometric isometry groups. The theory is burgeoning at the hands of Gromov, Epstein, Thurston, Holt, Bestvina, Feighn, Mess, Paterson, Gersten, Short, Shapiro, Parry, Floyd, Baumslag, Casson, Brick, Stallings, a whole school of continental mathematicians, Cannon, and others. The theory has produced the notions of automatic group, negatively curved group, complex of groups, almost convex groups, combable groups, isoperimetric and isodiametric inequalities for groups. Amid these many topics, Cannon, his students, and coworkers are currently concentrating on a cluster of projects related to the existence of hyperbolic structures on 3- manifolds. Is the generic finitely presented group negatively curved, as claimed without explicit proof by Gromov? Does every negatively curved closed 3-manifold admit a Riemannian metric of constant negative curvature? Can negative curvature be recognized algorithmically? Can the space at infinity be recognized algorithmically? Can potential constant curvature be recognized algorithmically? Cannon is using as the basis for these studies his combinatorial Riemann mapping theorem developed during the previous projects, the related work by Parry, and the computer Programs of Epstein and his collaborators. Groups are the appropriate algebraic structures for describing the notion of symmetry with precision. For this reason they play an extensive role in geometry (and topology). While the more usual interaction between group theory and geometry is the use of algebraic computations involving groups to prove theorems about geometric objects possessing symmetry, there is a very significant mathematical subdiscipline in which the roles are reversed and our intuition about geometric objects assists us in conjecturing and proving theorems about associated groups. A fascinating recent development is the influence of the computer in all this. Not only is it the ubiquitous timesaver that everyone knows, enabling one to undertake computations that would otherwise be too formidable, but it has actually influenced the questions in geometric group theory in a more fundamental way. Certain classes of groups have been defined in terms of the types of hypothetical machines that would be able to perform certain computations about them. Real hardware is not involved in this, and the same definitions could have been conceived long ago, but they were not, for the existence of computing machines has influenced the way we look at the world, the types of questions we ask about it, the kinds of properties that we find interesting.