ABSTRACT
Let K be a field of characteristic zero, let P1, . . . , Pk be distinct K-rational points of P1, and let C1, . . . , Ck be a family of conjugacy classes of a finite group G with trivial center. The following result is due to Belyi, Fried, Matzat, Shih, and Thompson (see [Ma23], [MM] and [Se8] for references).
Theorem 8.1.1 Assume that the family (C1, . . . , Ck) is rigid and that each Ci is rational. Then there is a regular G-covering C −→ P1 defined over K which is unramified outside {P1, . . . , Pk} and such that the inertia group over each Pi is generated by an element of Ci. Furthermore, such a covering is unique, up to a unique G-isomorphism.
