Mathematical Sciences: Subgroups of Residually Torsion- Free Nilpotent Metabelian Groups

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Research results over the last thirty years have indicated that finitely presented groups are very complex in their subgroup structure. For instance, it is known that a finitely generated metabelian group is a subgroup of a finitely presented metabelian group. In order to examine still further the complexity of finitely presented groups, it is of interest to determine whether a finitely generated metabelian group with a specific property can be embedded in a finitely presented metabelian group with the same property. The initial objective of the investigator had been to prove the general statement that if G is a finitely generated residually nilpotent (or residually torsion-free nilpotent) metabelian group, then G is a subgroup of a finitely presented metabelian group that is also residually nilpotent (or residually torsion-free nilpotent). This appears to be a complicated problem, as evidenced by the machinery required to solve it for the free metabelian case. In the special case of free metabelian groups it is known that finitely generated free metabelian groups are residually nilpotent and residually torsion-free nilpotent. The investigator has proved that a finitely generated free metabelian group can be embedded in a finitely presented metabelian group that is also residually nilpotent. Furthermore, a finitely generated free metabelian group can be embedded in a finitely presented metabelian group that is also residually torsion-free nilpotent. One of the tools developed by the investigator to obtain these results is a sufficient condition for the residual nilpotence and residual torsion-free nilpotence of certain finitely presented metabelian groups that arise from a matrix representation for metabelian groups. The matrix entries belong to specified groups and specified modules of fractions. To show the sufficient condition can be satisfied, use is made of the notion of S-isolator of an ideal, where S is a multiplicative semigroup. The investigat or is attempting to develop further insight and additional tools with which to try to solve the initial problem mentioned above. An extension to be considered is to let the matrix entries belong to certain factor groups of modules and to determine whether the resulting matrix group is residually torsion-free nilpotent.