Hurwitz action and finite quotients of braid groups.

*(English)*Zbl 0701.20019
Braids, AMS-IMS-SIAM Jt. Summer Res. Conf., Santa Cruz/Calif. 1986, Contemp. Math. 78, 299-325 (1988).

Summary: [For the entire collection see Zbl 0651.00010.]

In 1891, A. Hurwitz described an operation of braid groups on certain sets of Riemann surfaces. This operation naturally generalizes to interesting problems about mapping class groups of surfaces, integral quadratic forms and singularities. On the other hand, group actions always induce permutation representations of the group in question. The aim of this paper is to determine explicitly some of the braid group representations associated to Hurwitz’s operation.

In the first part, a description of the orbits will be given (Theorems 1 and 2). In the second section we will associate a diagram to each Riemann surface. This procedure will leave us with a class of diagrams, which is characterized in graph-theoretic terms by theorem 3. It turns out that the diagrams are a lot easier to handle than the Riemann surfaces. So we are able to show in theorem 6 that the braid groups operate as full symmetric or alternating groups on some sets of Riemann surfaces with at least five sheets over \(S^ 2\). For 3-sheeted surfaces the corresponding braid group representations are by symplectic groups over the field of three elements. In this case, by work of Birman and Wajnryb, isotropy groups and kernel of the representations are explicitly known (Theorems 7 and 8).

In 1891, A. Hurwitz described an operation of braid groups on certain sets of Riemann surfaces. This operation naturally generalizes to interesting problems about mapping class groups of surfaces, integral quadratic forms and singularities. On the other hand, group actions always induce permutation representations of the group in question. The aim of this paper is to determine explicitly some of the braid group representations associated to Hurwitz’s operation.

In the first part, a description of the orbits will be given (Theorems 1 and 2). In the second section we will associate a diagram to each Riemann surface. This procedure will leave us with a class of diagrams, which is characterized in graph-theoretic terms by theorem 3. It turns out that the diagrams are a lot easier to handle than the Riemann surfaces. So we are able to show in theorem 6 that the braid groups operate as full symmetric or alternating groups on some sets of Riemann surfaces with at least five sheets over \(S^ 2\). For 3-sheeted surfaces the corresponding braid group representations are by symplectic groups over the field of three elements. In this case, by work of Birman and Wajnryb, isotropy groups and kernel of the representations are explicitly known (Theorems 7 and 8).

##### MSC:

20F36 | Braid groups; Artin groups |

05C25 | Graphs and abstract algebra (groups, rings, fields, etc.) |

30F10 | Compact Riemann surfaces and uniformization |

57N05 | Topology of the Euclidean \(2\)-space, \(2\)-manifolds (MSC2010) |