Mathematical Sciences: Research in Geometric Group Theory

在过去的几年中，几何群论在词双曲群、自动群和非正曲率空间等领域出现了新的方向。查尼教授和戴维斯教授将继续研究这些领域的一些问题。查尼（与夏皮罗先生一起）计划研究与Artin群体的自动性有关的几个问题。她还计划研究在几个类似建筑的建筑群上使用非正曲率度量的问题。戴维斯（也和夏皮罗一起）计划研究所有考克斯特组都是自动的这一猜想。与此相关的是他研究Coxeter群边界性质的项目。这在寻找上同调维2但几何维3的一组（Eilenberg-Ganea问题）时可能也很有用。最后，Charney和Davis打算共同研究严格夸张化程序和相对夸张化程序。“严格”是指将复合体转换为严格负曲率空间（而不仅仅是非正曲率）的可能性。“相对”指的是保持子复上的度规固定的问题。查尼和戴维斯还计划继续他们对非正弯曲轨道的研究。这里的问题是确定球体上的某些分段球度量何时是“大”的。群是精确描述对称概念的适当代数结构。由于这个原因，它们在几何（和拓扑）中扮演着广泛的角色。虽然群论和几何之间更常见的相互作用是使用涉及群的代数计算来证明关于具有对称性的几何物体的定理，但在一个非常重要的数学分支学科中，角色是相反的，我们对几何物体的直觉帮助我们推测和证明关于相关群的定理。最近一项引人入胜的发展是计算机对这一切的影响。它不仅是大家都知道的无处不在的省时工具，使人们能够进行原本过于艰巨的计算，而且它实际上以一种更根本的方式影响了几何群论中的问题。某些类别的群体已经根据假想的机器的类型来定义，这些机器将能够对它们进行某些计算。真正的硬件与此无关，同样的定义在很久以前就可以被构想出来，但它们没有，因为计算机器的存在影响了我们看待世界的方式，影响了我们对世界提出的问题类型，影响了我们感兴趣的属性类型。

New directions in geometric group theory have emerged in the past few years in the areas of word hyperbolic groups, automatic groups, and spaces of nonpositive curvature. Professors Charney and Davis will continue research on a number of problems in these areas. Charney (together with M. Shapiro) plans research on several questions related to the automaticity of Artin groups. She also plans to study the problem of putting a metric of nonpositive curvature on several building-like complexes. Davis (also with M. Shapiro) plans to work on the conjecture that all Coxeter groups are automatic. Related to this is his project to study properties of the boundary of a Coxeter group. This might also be useful in finding a group of cohomological dimension two but geometric dimension three (the Eilenberg-Ganea Question). Finally, jointly, Charney and Davis intend to work on strict hyperbolization procedures and relative hyperbolization procedures. "Strict" refers to the possibility of converting a complex into a space of strictly negative curvature (rather than just nonpositive curvature). "Relative" refers to the problem of keeping the metric on a subcomplex fixed. Charney and Davis also plan to continue their research on nonpositively curved orbifolds. The issue here is to decide when certain piecewise spherical metrics on spheres are "large." 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.