论文标题
自动群体理论的发展
The development of the theory of automatic groups
论文作者
论文摘要
我们描述了自动群体理论的发展。我们从历史介绍开始，定义自动，双重和混合组的概念，得出基本特性，然后解释如何基于瑟斯顿的八个几何形状中的六个基于透明的3个序列组和紧凑型3个manifolds组。我们描述了沃里克开发的软件，以计算自动结构以及使用这些结构的实用算法的开发。我们解释了如何使用各种表现出负曲率概念的空间的群体的行为来证明自动性或双重性，并展示这些结果如何用于在某些无限家庭（辫子组，映射群体，Artin组的家族，Artin组和Coxeter组）中为组得出这些特性。在整个文本中，我们提出了一段时间但现在已经解决的问题，这些问题和问题保持开放。
We describe the development of the theory of automatic groups. We begin with a historical introduction, define the concepts of automatic, biautomatic and combable groups, derive basic properties, then explain how hyperbolic groups and the groups of compact 3-manifolds based on six of Thurston's eight geometries can be proved automatic. We describe software developed in Warwick to compute automatic structures, as well as the development of practical algorithms that use those structures. We explain how actions of groups on spaces displaying various notions of negative curvature can be used to prove automaticity or biautomaticity, and show how these results have been used to derive these properties for groups in some infinite families (braid groups, mapping class groups, families of Artin groups, and Coxeter groups). Throughout the text we flag up open problems as well as problems that remained open for some time but have now been resolved.
