Abstract. The study of one-relator groups includes the connections between group properties and the form of the relator. In this paper we discuss conditions on the form w1v'uvm which force the corresponding one-relator groups to be nonresidually finite, i.e. the intersection of the normal subgroups of finite index to be nontrivial. Moreover we show that these forms can be detected amongst the words of a free group.

Abstract. The study of one-relator groups includes the connections between group properties and the form of the relator. In this paper we discuss conditions on the form w1v'uvm which force the corresponding one-relator groups to be nonresidually finite, i.e. the intersection of the normal subgroups of finite index to be nontrivial. Moreover we show that these forms can be detected amongst the words of a free group.

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Communicated by M. F. Newman ABSTRACT. Many one-relator groups with center have been shown to be of the form -A necessary and a sufficient condition for the sequence (Pi,Qi,P 2 >Q2> '",Pt>Qt) a r e given in order for groups of the above form to be one-relator groups.

Abstract.The group described in the title is obtained as a quotient of a center-by-metabelian group constructed by P. Hall.It is well known that a residually finite group which is finitely presented has a solvable word problem (V. H. [5, p. 307]) that a residually finite group which is only finitely generated may have an unsolvable word problem. Herein we exhibit such a group which moreover is soluble, indeed it is center-by-metabelian. This is in some sense a minimal soluble example since all finitely generated metabelian groups have a solvable word problem. (This is an easy consequence of the fact that finitely generated free metabelian groups satisfy the maximal condition on normal subgroups (P. Hall [2]) and thus any finitely generated metabelian group can be almost finitely presented in the sense of Mostowski [6] and is residually finite (P. Hall [3]).)Credit must be given to Peter Neumann who pointed out the appropriate example to be modified.Our example uses a recursive function / with a nonrecursive range in a manner similar to the way Higman [5] exhibited a finitely presented group with an unsolvable word problem.Let G be the center-by-metabelian group constructed by P. Hall in ([2, p. 434]). In the notation of P. Hall, G is generated by a and b and defined by