Decidability problems in automaton semigroups

TL;DR

This paper investigates the decidability of various problems in automaton semigroups, providing algorithms for the word problem, Engel identities, and recognizing Engel elements, with specific results on Grigorchuk's group.

Contribution

It introduces new algorithms for the word problem and Engel identity decision problems in automaton semigroups, including Grigorchuk's group, and establishes their decidability in specific cases.

Findings

Decidable algorithms for the word problem in automaton semigroups.

A partial algorithm for Engel identities in automaton groups.

Decidability of Engel elements in Grigorchuk's group, characterized by order at most 2.

Abstract

We consider decidability problems in self-similar semigroups, and in particular in semigroups of automatic transformations of $X^*$. We describe algorithms answering the word problem, and bound its complexity under some additional assumptions. We give a partial algorithm that decides in a group generated by an automaton, given $x,y$, whether an Engel identity ($[\cdots[[x,y],y],\dots,y]=1$ for a long enough commutator sequence) is satisfied. This algorithm 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 $2$. We include, in the text, a large number of open problems. Our computations were implemented using the package "Fr" within the computer algebra system "Gap".