Translator’s Introduction 5

Abstract

The next paper contains Dehn’s last word on the conjugacy problem for surface groups, as well as a very simple solution of the word problem. The solutions, by what is known as Dehn’s algorithm, are based on a detailed combinatorial study of closed paths in the group diagram. This study shows, in effect, that if a word for an element of the fundamental group can be reduced in length, then it can be reduced monotonically, namely by replacing any subword which is more than half the defining relator by its complement, and by trivial cancellations. (When only a conjugate of the original element is sought, the word is regarded as circular in applying these operations.)