An automaton group with undecidable order and Engel problems

TL;DR

This paper constructs a specific automaton group that encodes Turing machine behavior, demonstrating that the group has undecidable order and Engel problems, thus solving longstanding open problems in group theory.

Contribution

It explicitly constructs automaton groups with undecidable order and Engel problems, linking group properties to Turing machine universality.

Findings

Automaton groups can simulate Turing machines.

Decidability of order and Engel problems can be undecidable in automaton groups.

Solves open problems posed by Grigorchuk and Bartholdi.

Abstract

For every Turing machine, we construct an automaton group that simulates it. Precisely, starting from an initial configuration of the Turing machine, we explicitly construct an element of the group such that the Turing machine stops if, and only if, this element is of finite order.If the Turing machine is universal, the corresponding automaton group has an undecidable order problem. This solves a problem raised by Grigorchuk.The above group also has an undecidable Engel problem: there is no algorithm that, given g, h in the group, decides whether there exists an integer n such that the n-iterated commutator [...[[g,h],h],...,h]$ is the identity or not. This solves a problem raised by Bartholdi.