Project:Sandbox

版权归作者所有，任何形式转载请联系作者.

作者：家禽腿部保健（来自豆瓣）

来源： https://www.douban.com/note/590688874/

2016年第三届濑户内海艺术祭在11月五日已经结束，我赶上了最后几天. 这三年一度的艺术祭，我可是从春天盼到了夏天，又从夏天盼到了秋天，秋会期再不去就要再等三年，三年啊！我可不想三年后可能拖家带口再去一个个看，终于在出发前三天才买了春秋航空上海到高松的机票，一直待到回来前五天才买回程机票，竟然含税才一千元.

这回待了15天，除了三天跑到松山去看看外，其他时间几乎都待在高松，在各小岛上待着，散漫着. 高松港是往艺术祭各小岛展区的出发点，亦是濑户内海轮渡的母港，船班进进出出，游客每天都从这里出发

Groups and geometry are ubiquitous in mathematics, groups because the symmetries (or automorphisms [I.3 §4.1]) of any mathematical object in any context form a group and geometry because it allows one to think intuitively about abstract problems and to organize families of objects into spaces from which one may gain some global insight.

The purpose of this article is to introduce the reader to the study of infinite, discrete groups. I shall discuss both the combinatorial approach to the subject that held sway for much of the twentieth century and the more geometric perspective that has led to an enormous flowering of the subject in the last twenty years. I hope to convince the reader that the study of groups is a concern for all of mathematics rather than something that belongs particularly to the domain of algebra.

The principal focus of geometric group theory is the interaction of geometry/topology and group theory, through group actions and through suitable translations of geometric concepts into group theory. One wants to develop and exploit this interaction for the benefit of both geometry/topology and group theory. And, in keeping with our assertion that groups are important throughout mathematics, one hopes to illuminate and solve problems from elsewhere in mathematics by encoding them as problems in group theory.

Geometric group theory acquired a distinct identity in the late 1980s but many of its principal ideas have their roots in the end of the nineteenth century. At that time, low-dimensional topology and combinatorial group theory emerged entwined. Roughly speaking, combinatorial group theory is the study of groups defined in terms of presentations, that is, by means of generators and relations. In order to follow the rest of this introduction the reader must first understand what these terms mean. Since their definitions would require an unacceptably long break in the flow of our discussion, I will postpone them to the next section, but I strongly advise the reader who is unfamiliar with the meaning of the expression Γ = < a1, ..., an | r1, ..., rm > to pause and read that section before continuing with this one.