The characterization by automata of certain profinite groups

We use the combinatorial language of automata to define and study profinite groups which are infinitely iterated permutational wreath products of transitive finite permutation groups. We provide some naturally defined automaton realizations of these groups by the so-called time-varying automata, as well as by Mealy automata to characterize these groups as automaton groups, i.e. as groups topologically generated by a single automaton.