##
**Automorphic sets and braids and singularities.**
*(English)*
Zbl 0716.20017

Braids, AMS-IMS-SIAM Jt. Summer Res. Conf., Santa Cruz/Calif. 1986, Contemp. Math. 78, 45-115 (1988).

[For the entire collection see Zbl 0651.00010.]

The contents of the survey is as follows. §1 is a brief introduction to some of Artin’s classical results on braid groups, presented in a way fit for future developments. §2 introduces the automorphic sets together with a few categorical definitions obviously related to this notion and gives many examples. §3 deals with the operation of \(B_ n\) and certain other groups on the cartesian products \(\Delta^ n\) of automorphic sets \(\Delta\) and introduces invariants of the orbits, in particular pseudo Coxeter elements, Coxeter diagrams and monodromy groups. This is followed by a discussion of problems related to the operation of \(B_ n\) on \(\Delta^ n\) and a report on classical and recent work on these problems. Much of this is related to root systems \(\Delta\), which are a particularly nice class of automorphic sets. §4 finally explains the applications to singularities.

The contents of the survey is as follows. §1 is a brief introduction to some of Artin’s classical results on braid groups, presented in a way fit for future developments. §2 introduces the automorphic sets together with a few categorical definitions obviously related to this notion and gives many examples. §3 deals with the operation of \(B_ n\) and certain other groups on the cartesian products \(\Delta^ n\) of automorphic sets \(\Delta\) and introduces invariants of the orbits, in particular pseudo Coxeter elements, Coxeter diagrams and monodromy groups. This is followed by a discussion of problems related to the operation of \(B_ n\) on \(\Delta^ n\) and a report on classical and recent work on these problems. Much of this is related to root systems \(\Delta\), which are a particularly nice class of automorphic sets. §4 finally explains the applications to singularities.

### MSC:

20F36 | Braid groups; Artin groups |

32S30 | Deformations of complex singularities; vanishing cycles |

14B05 | Singularities in algebraic geometry |

20B27 | Infinite automorphism groups |

17B20 | Simple, semisimple, reductive (super)algebras |

32S40 | Monodromy; relations with differential equations and \(D\)-modules (complex-analytic aspects) |