On some universal construction of minimal topological generating sets for inverse limits of iterated wreath products of non-Abelian finite simple groups

Let (Formula Presented.) be an arbitrary sequence of non-Abelian finite simple transitive permutation groups. By using the combinatorial language of time-varying automata, we provide an explicit and naturally defined construction of a two-element set which generates a dense subgroup in the inverse limit (Formula Presented.) of iterated permutational wreath products of the groups H_i. The corresponding automaton is equipped with three states, one of which is neutral and the semigroup generated by the other two states is free. We derive other algebraic and geometric properties of the group generated by this automaton. By using the notion of a Mealy automaton, we obtain the analogous construction for the infinite permutational wreath power of an arbitrary non-Abelian finite simple transitive permutation group H on a set X. We show that the wreath power (Formula Presented.) contains a dense 2-generated not finitely presented amenable subgroup of exponential growth, which is generated by a 3-state Mealy automaton over the alphabet X. The self-similar group generated by this automaton is self-replicating, contracting and regular weakly branch over the commutator subgroup.