Algorithmic decidability of Engel's property for automaton groups

TL;DR

This paper investigates the decidability of Engel's property in automaton groups, providing algorithms that determine whether elements satisfy Engel identities, with specific results for Grigorchuk's group.

Contribution

It introduces a partial algorithm for Engel identity decision problems in automaton groups and characterizes Engel elements in Grigorchuk's group.

Findings

Decidability of Engel identity for certain automaton groups.

Grigorchuk's 2-group is proven not to be Engel.

Engel elements in Grigorchuk's group are exactly those of order at most 2.

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.