Algorithmic Decidability of Engel's Property for Automaton Groups

We consider decidability problems associated with Engel’s identity ([· · · [[x, y], y], . . . , y] = 1 for a long enough commutator sequence) in groups generated by an automaton. We give a partial algorithm that decides, given x, y, whether an Engel identity is satisfied. It succeeds, importantly, in proving that Grigorchuk’s 2-group is not Engel. We consider next the problem of recognizing Engel elements, namely elements y such that the map x 7→ [x, y] attracts to {1}. Although this problem seems intractable in general, we prove that it is decidable for Grigorchuk’s group: Engel elements are precisely those of order at most 2. Our computations were implemented using the package Fr within the computer algebra system Gap.