In the 1970s, Joan Birman and Hugh Hilden wrote several papers on the problem of relating the mapping class group of a surface to that of a covering space. Their results provide a bridge between the theories of mapping class groups and braid groups. We survey the work of Birman and Hilden, give an overview of the subsequent developments, and discuss open questions and new directions.

中文翻译：

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辫子组和映射类组：Birman-Hilden 理论

在 1970 年代，Joan Birman 和 Hugh Hilden 写了几篇关于将表面的映射类群与覆盖空间的映射类群相关联的问题的论文。他们的结果在映射类组和辫子组的理论之间架起了一座桥梁。我们调查了伯曼和希尔登的工作，概述了随后的发展，并讨论了开放性问题和新方向。