Reflections on the residual finiteness of one-relator groups

Abstract

Let G = < a, b, … | r = 1 > be a one-relator group equipped with at least two generators. For all w which do not commute with r in the ambient free group on the generators a, b, …, the groups G(r,w) = < a,b, … | r r w = r 2 > are not residually finite and have the same finite images as G. The existence of this family of one-relator groups which are not residually finite reinforces what is becoming more obvious with time, that one-relator groups can be extremely complicated. This not only serves to underline the complexity of one-relator groups but provides us with the opportunity to raise a number of problems about these groups in the hope that they will stimulate further work on the conjugacy and isomorphism problems for one-relator groups as a whole.