14 - Finiteness conditions and the word problem

Summary

The purpose of this note is a generalization of the solution of the word problem for recursively presented simple groups. The results are pretty trivial but, like other easy observations in this field, they may have useful applications. Most of the work is accomplished by the choice of an appropriate terminology, some of which has been introduced in, and. Most ordinary decision algorithms consist of a reduction procedure to a normal form that yields the desired decision by a directly discernible typographical property. Thus triviality is reduced to zero length in free groups and the solutions of the word problem for individual finite or nilpotent groups are also of this kind. A uniform solution for a class Ḵ of groups extends to a solution for finitely presented residually-Ḵ-groups. That finite relatedness is indispensible here, is shown by examples of finitely generated, recursively related, residually finite groups with unsolvable word problems. On the other hand there are structural features of the lattice of normal subgroups that, in conjunction with a recursive presentation, will yield a decision algorithm for the word problem, while the class of all finite presentations with that property is not recursively enumerable. Finiteness of the normal lattice is a typical example and so is the property of being nearly critical in a sense that will be made precise below.