Residual Nilpotence of Groups with One Defining Relation

Abstract

All groups in the family of Baumslag-Solitar groups (i.e., groups of the form G(m,n) = 〈a, b; a−1bma = bn〉, where m and n are nonzero integers) for which the residual nilpotence condition holds if and only if the residual p-finiteness condition holds for some prime number p are described. It has turned out, in particular, that the group G(pr, −pr), where p is an odd prime and r ≥ 1, is residually nilpotent, but it is residually q-finite for no prime q. Thus, an answer to the existence problem for noncyclic one-relator groups possessing such a property (formulated by McCarron in his 1996 paper) is obtained. A simple proof of the statement that an arbitrary residually nilpotent noncyclic one-relator group which has elements of finite order is residual p-finite for some prime p, which was announced in the same paper of McCarron, is also given.