Topological Properties of Groups and Group Extensions

Abstract

In Chapter I.5 we described the first productive uses made by Dehn and others of Cayley’s discovery of the graph of a group. In (rather vague) general terms, this may be described as a method of associating a topological cell complex (in the case of the graph, a 1-complex) with the group and deriving results about the group from this association. The first progress beyond the methods used by Dehn is probably the connection between covering spaces and subgroups of the fundamental group of the base space which had been utilized by Reidemeister [1932b]. An important and very productive generalization was introduced by van Kampen [1933a and 1933b], who uses 2-complexes. His ideas, which became fully effective only 30 years later, have been explained by Lyndon and Schupp [1977, pp. 151 and 236]. Chapters 3 and 5 of their work deal with these geometric methods, explaining all technicalities and providing some historical information. The title of Chapter 5 is Small cancellation theory. Lyndon and Schupp treat it by geometric methods. It started as a purely algebraic theory in a paper by Tartakovskii [1949], who provided algorithms for the solution of word problems in a large class of groups which have presentations in which only sufficiently small parts of the relators can cancel each other. These algorithms are, in part, highly complex.