Translator’s Introduction 2

Abstract

The following chapter of Dehn’s lecture notes deals mainly with surface topology, presenting several of the results which were eventually published in Dehn [1912a]. The account here is different however, giving more details on non-orientable surfaces, and giving different proofs of his main result, the solution of the conjugacy problem (or “transformation problem” as Dehn calls it). These proofs greatly illuminate the curious solution given in Dehn [1912a], which is a combinatorial algorithm but justified by appeal to the hyperbolic metric, revealing it as a transitional stage between the metric solutions and the purely combinatorial solution given in Dehn [1912b].