Translator’s Introduction 4

Abstract

This famous and influential paper is a memorable introduction to topology, combinatorial group theory and noneuclidean geometry. To see Dehn weave these threads into a coherent pattern is an unforgettable demonstration of the unity of mathematics. Of course, there is a historical basis for this coherence in the work of Klein and Poincaré in the 1880’s, and the relation between surface topology and hyperbolic geometry had already been used with considerable sophistication by Poincaré [1904], but Dehn’s paper is the first, I believe, to show equal sophistication in the treatment of combinatorial group theory. In particular, it is the first in which a purely combinatorial algorithm is justified by appeal to the hyperbolic metric.