On a Class of One-Relator Groups

Abstract

In this paper, we consider the class of groups G(l, m; k) which are defined by the presentationwhere k, l, m are integers, and |l| > m > 0, k > 0. Groups in this class possess many properties which seem unusual, especially for one-relator groups. The basis for the results obtained below is the determination of endomorphisms.For certain of the groups, we are able to calculate their automorphism groups. One consequence of this is to produce examples of one-relator groups with infinitely generated automorphism groups. This answers a question raised by G. Baumslag (in a colloquium lecture at the University of Waterloo). Our examples are, perhaps, the simplest possible; J. Lewin [10] has found an example of a finitely presented group with an infinitely generated automorphism group.