Paper Info
Title
Algorithmic decidability of Engel's property for automaton groups
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,ï¾źy, 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.
Year
DOI
Venue
2015
10.1007/978-3-319-34171-2_3
CSR
Field
DocType
Volume
Discrete mathematics,Combinatorics,Automaton,Symbolic computation,Decidability,Commutator (electric),Mathematics,Computation
Journal
abs/1512.01717
ISSN
Citations 
PageRank 
CSR 2016, Springer LNCS 9691 (2016) 29--40
0
0.34
References 
Authors
2
1
Name
Order
Citations
PageRank
Laurent Bartholdi1278.74