Finite generation of iterated wreath products in product action

Abstract

Let \({\mathcal{S}}\) 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 \({\mathcal{S}}\) is topologically finitely generated, provided that the actions of the groups in \({\mathcal{S}}\) are never regular. We also deduce that certain infinitely iterated wreath products obtained by a mixture of imprimitive and product actions of groups in \({\mathcal{S}}\) are finitely generated. Finally we apply our methods to find explicitly two generators of infinitely iterated wreath products in product action of certain sequences \({\mathcal{S}}\) of 2-generated perfect groups.