On the Nielsen Equivalence of Pairs of Generators in Certain HNN Groups

Abstract

This paper is concerned with obtaining information about the Nielsen equivalence classes and T-systems of certain two-generator HNN groups. The principal result (Theorem 1) states that the one-relator groups $$\left\langle {a,t;{{\left[ {{a^{{a_1}}}{t^{ - 1}}{a^{\varepsilon {\beta _1}}}t \ldots {a^{{\alpha _r}}}{t^{ - 1}}{a^{\varepsilon {\beta _n}}}t} \right]}^n}} \right\rangle \left( {r > 0} \right)$$ where α i , β i (i = 1, 2, ..., r) are positive integers, |e| = 1, n > 1, have one Nielsen equivalence class of generating pairs. As a corollary of this result a counterexample to the converse of Corollary 4.13.1 of Magnus, Karrass and Solitar, “Combinatorial Group Theory” is obtained. The other main result of the paper (Theorem 2) gives a fairly detailed description of the Nielsen equivalence classes and T-systems of some HNN extensions of certain small cancellation groups.