Finite generation of iterated wreath products in product action

摘要

Let S be a sequence of finite perfect transitive permutation groups with uniformly bounded number of generators. We prove that the infinitely iterated wreath product in product action of the groups in is topologically finitely generated, provided that the actions of the groups in are never regular. We also deduce that certain infinitely iterated wreath products obtained by a mixture of imprimitive and product actions of groups in are finitely generated. Finally we apply our methods to find explicitly two generators of infinitely iterated wreath products in product action of certain sequences of 2-generated perfect groups.