One-Relator Groups

Abstract

According to a theorem mentioned at the end of Chapter II.4, every finitely presented group can be embedded in a two-generator group with the same number of defining relations. This shows that, at least for the word problem, the number of generators of a group is immaterial if it is at least two. Intuitively, this fact must have been known almost from the beginning of the theory of groups given by presentations. It is the defining relations which make even the word problem so difficult. Indeed, for free (i.e., for no-relator) groups, the solution of the word problem was at least intuitively obvious already to Dyck in 1882. But for nearly half a century, the word problem for one-relator groups had been solved mainly with geometric methods for some knot groups and for fundamental groups of two-dimensional manifolds until Schreier [1927a] observed that his theorem on free products with amalgamations permits its solution for the class of one-relator groups mentioned in Chapter II.4. This paper contains practically all of the earlier examples.