The word problem and power problem in 1-relator groups are primitive recursive

Abstract

The purpose of this paper is to show the solution to the word problem in a 1-relator group can be computed with respect to an effective indexing of the group by an algorithm at level at most 2+(R) of the Grzegorczyk hierarchy, where (R) is the length of the relator, and by a primitive recursive function, always. As a consequence, it is shown that the power problem in a 1-relator group can be solved similarly. An example is given in which the Magnus algorithm for the extended word problem is at level 4 but not 3 of the Grzegorczyk hierarchy even though the word problem is solvable at level 3.