On the generation of one-relator groups

Abstract

This paper is concerned with obtaining information about the Nielsen equivalence classes and T T -systems of certain two-generator HNN groups, and in particular of certain two-generator one-relator groups. The theorems presented here extend results of the author appearing in the Proceedings of the Second International Conference on the Theory of Groups. In particular it is shown here that if G = ⟨ a , t ; a α 1 E r − 1 a β 1 E r ⋯ a a s E r − 1 a B s E r ) G = \langle a,t;{a^{{\alpha _1}}}E_r^{ - 1}{a^{{\beta _1}}}{E_r} \cdots {a^{{a_s}}}E_r^{ - 1}{a^{{B_s}}}{E_r}) where the α j {\alpha _j} are positive, the β i {\beta _i} are nonzero, E r {E_r} has the form [ a ∈ 1 , [ a ∈ 2 , [ ⋯ , [ a ∈ r , t ] ⋯ ] ] ] [{a^{{ \in _1}}},[{a^{{ \in _2}}},[ \cdots ,[{a^{{ \in _r}}},t] \cdots ]]] with | ∈ 1 | = | ∈ 2 | = ⋯ = | ∈ r | = 1 |{ \in _1}| = |{ \in _2}| = \cdots = |{ \in _r}| = 1 , then in a large number of cases G G has one Nielsen equivalence class. Similar results are also obtained for certain groups with more than one relator. A fair proportion of the paper is given to developing a method for reducing pairs of elements in HNN groups. This method has some of the features of Nielsen’s reduction theorem for free groups. One other interesting result obtained here is that a one-relator group with torsion which has one T T -system is Hopfian. The early part of the paper is discursive. It contains most of the known results concerning T T -systems of one-relator groups, and highlights several open problems, some of which have been raised by other authors.