The word and order problems for self-similar and automata groups

TL;DR

This paper proves that certain computational problems, specifically the word and order problems, are undecidable in classes of self-similar and automata groups, highlighting fundamental limits in their algorithmic analysis.

Contribution

It establishes the undecidability of the word problem in functionally recursive groups and the order problem in automata groups, even when these groups are contracting.

Findings

Word problem is undecidable in functionally recursive groups.

Order problem is undecidable in automata groups.

Undecidability holds even for contracting automata groups.

Abstract

We prove that the word problem is undecidable in functionally recursive groups, and that the order problem is undecidable in automata groups, even under the assumption that they are contracting.