On the word problem for just infinite groups

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.