Submission date: 6 December 2015

Abstract:

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 → [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.

Mathematics Subject Classification:

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Full text arXiv 1512.01717: pdf, ps.