On free products with amalgamation of two infinite cyclic groups

On the Generation of One-Relator Groups

“…An example is provided by the group G = <Xj, x2;x2 = x2 >, which actually has infinitely many T-systems [24], [6]. It should also be pointed out that, contrary to a conjecture of Magnus, not every one-relator group satisfies the assumptions of Theorem l(i) ( [4], [13], and §1.3 below).…”

“…The previous subsection shows that a determination of the Nielsen equivalence classes and T-systems of a group, particularly a one-relator group, is of considerable importance, and this problem has received the attention of several authors (see, for example, [4] - [6], [13], [16], [18], [19], [22] - [26] ). In this subsection a theorem is obtained which will be used in §1.3 below to show that certain two-generator onerelator groups with torsion have one Nielsen equivalence class.…”

On Recognising Certain One Relation Presentations

“…Let k, I be integers different from 0, ± 1. It was shown in [2] that if r, s are integers satisfying (r, s) = (r, k) = (s, l)=l, then the group (xx, x2; x\=xl2) has presentation 11 == \X^, -^2» XiXqXi X% --1 , X-^XqXi X% ^= 1 , X-± X% ^-1/.…”

“…In the study of these groups, the problem arises of determining for which values of/-and y the presentation If is (in the notation of [2]) a one relation presentation. The object of this note is to prove a result which has as a consequence the fact that, given k, I, r and s, it can be determined if the corresponding presentation II is a one relation presentation.…”

“…From n we can obtain, by Theorem 3.5 of [1], a presentation Il1 = {b1,bi;V1=l,---,Vn = l) of G which is N-equivalent (in the notation of [2]) to n, and such that…”