Max Dehn’s Contribution To Combinatorial Group Theory

The combinatorial group theory is a rather young frontier, Dehn who is the student ofHilbert has made significant contribution to the subject, throughout the entire developmentprocess, such as sprouting period, founding period and the period of independent development.Dehn creates the branch of mathematics-combinatorial group theory, by putting togother thethought of Hilbert and Poincaré who are two of the greatest mathematicians in the lateNineteenth Century and early Twentieth Century. Dehn’s work in the1910s put forward theimportant method which was called the group diagram and Dehn’s problem which has beenstimulating the development of the subject.Simultaneously, all these stand for the originatingof the discipline.There are crosses between combinatorial group theory and many other branches ofmathematics, especially geometry and topology. The history of combinatorial group theory inthis period is exactly the result of interdisciplinary development. This dissertation which isbased on full possession of documents analyzes and studies the prehistory of the combinationof group theory, Dehn’s contributions and profound implications. The main results are asfollows:1This dissertation summarizes the prehistory of the combinatorial group theory, which isknown as the theory of presentation. With the development of abstraction in group theory, thediscrete groups issuing from grometry and the theory of automorphic functions give birth tothe conception of presentation. This paper not only analyzes the originating of the conceptionand its topological applications, but also tries to elaborate the contribution of Dehn to thesubject.2This dissertation explores the history of a geometric method—the group diagram, andcompares it with the similar conception put forword by other mathematicians such as Cayleyand Burnside. The dissertation analyzes the whole story of Dehn’s problem,elabrates theprogress of Dehn’s problem in combinatorial group theory by using the group diagram. Theprogress not only includes the specific results, like the results coming from the study of the knot group, but also the method and theory, such as small cancellation theory. These clarifythe meaning of the Dehn’s work on the development of a combination of group theory.3This dissertation elaborates Dehn’s life. The great mathematician doesn’t get the attentionhe deserves under the aura of his tutor Hilbert. Throughout making his contributions to thecombinatorial group theoy as the starting point, the dissertation analyzes the research ideasand ideological course.4This dissertation studies the academic inheritance relationship between the successor inthe field, makes the status of Dehn’s and his students’ in the developing history of thecombinatorial group theory clear.