Abstract
We consider decidability problems associated with Engel’s identity (\([\cdots [[x,y],y],\dots ,y]=1\) 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 \(x\mapsto [x,y]\) 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.
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L. Bartholdi—Partially supported by ANR grant ANR-14-ACHN-0018-01.
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Acknowledgments
I am grateful to Anna Erschler for stimulating my interest in this question and for having suggested a computer approach to the problem, and to Ines Klimann and Matthieu Picantin for helpful discussions that have improved the presentation of this note.
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Bartholdi, L. (2016). Algorithmic Decidability of Engel’s Property for Automaton Groups. In: Kulikov, A., Woeginger, G. (eds) Computer Science – Theory and Applications. CSR 2016. Lecture Notes in Computer Science(), vol 9691. Springer, Cham. https://doi.org/10.1007/978-3-319-34171-2_3
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