Finite groups defined by presentations in which each defining relator involves exactly two generators

We consider two classes of groups, denoted JΓ and MΓ, defined by presentations in which each defining relator involves exactly two generators, and so are examples of simple Pride groups. (For MΓ the relators are Baumslag-Solitar relators.) These presentations are, in turn, defined in terms of a non-trivial, simple directed graph Γ whose arcs are labelled by integers. When Γ is a directed triangle the groups JΓ,MΓ coincide with groups considered by Johnson and by Mennicke, respectively. When the arc labels are all equal the groups form families of so-called digraph groups. We show that if Γ is a non-trivial, strongly connected tournament then the groups JΓ are finite, metabelian, of rank equal to the order of Γ and we show that the groups MΓ are finite and, subject to a condition on the arc labels, are of rank equal to the order of Γ.