On the automaton complexity of wreath powers of non-abelian finite simple groups

We show that for every n⩾5 the infinite permutational wreath power of the alternating group of degree n with its natural permutation representation is topologically generated by a 2-state automaton, answering the question on the existence of a minimal automaton realization for an infinite wreath power of a non-trivial group. We also extend this result to some 2-generated perfect groups. Finally, we show that every non-abelian finite simple group admits a faithful and transitive action on a finite set such that the corresponding wreath power has an almost minimal automaton realization, extending the result from [15].