The residual finiteness of a class of 1-relator groups

Abstract

In [3] Gilbert Baumslag asserted that, for non-zero integers α, β, γ, δ such that α + γ ≠ 0 ≠ β + δ, the group G = <a, b:aα, bβaγbδ> is residually finite (RF). This result has been quoted in the literature: for example, in [2]. At the “Groups '85” meeting at St. Andrews, the second author learned, indirectly, that Professor Baumslag could not recall all the details of the rather complicated (unpublished) proof he had constructed and that he referred those asking for a proof to the present authors. It thus seems worthwhile formally to record the following fairly short proof of the above claim.