Algorithmic Decidability of Engel’s Property for Automaton Groups

  • Laurent BartholdiEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9691)


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 xy, 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.



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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© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.École Normale SupérieureParisFrance
  2. 2.Georg-August-Universität zu GöttingenGöttingenGermany

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