On the word problem for just infinite groups

TL;DR

This paper investigates the decidability of the word problem in just infinite groups, establishing conditions under which it is decidable or undecidable, and providing new proofs without relying on existing classification theorems.

Contribution

It offers new results on the decidability of the word problem for finitely and countably generated just infinite groups, avoiding the Wilson–Grigorchuk theorem and using classical decidability ideas.

Findings

Word problem is uniformly decidable for finitely generated just infinite groups with recursively enumerable relations.

For countably generated presentations, the word problem is decidable except in certain locally finite groups.

Constructs examples of locally finite groups with undecidable word problem, yet others with decidable word problem.

Abstract

We establish several results on the word problem for just infinite groups. First, for finitely generated just infinite groups we show that the word problem is uniformly decidable for presentations with recursively enumerable sets of relations. Our proof does not use the Wilson--Grigorchuk theorem on the classification of just infinite groups and proceeds directly from the definition, using ideas from classical results on decidability of the word problem: Kuznetsov's theorem and McKinsey--Maltsev theorem. For countably generated presentations of just infinite groups with a recursively enumerable set of relations we show that the word problem is decidable in all cases except locally finite groups without a computable lower bound on subgroup sizes. Finally, we construct presentations of countably generated locally finite groups with recursively enumerable set of relations, for which the word problem is undecidable. Yet, there exist other presentations of these groups, for which the word problem is decidable.