Belyi's theorem revisited.
Block systems of a Galois group.
Braid group action, embedding problems and the groups PGLn(q), Pun(q2).
Branching data for algebraic functions and representability by radicals
The branching data of an algebraic function is a list of orders of local monodromies around branching points. We present branching data that ensure that the algebraic functions having them are representable by radicals. This paper is a review of recent work by the authors and of closely related classical work by Ritt.