Finite-Rank Free Metabelian Subgroups of Finitely Presented Metabelian Groups

AbstractA sufficient condition is obtained for the residual torsion-free nilpotence of certain finitely presented metabelian groups that arise from a matrix representation developed by Magnus (1939,Ann. of Math.40, 764–768) for metabelian groups. Using this condition and a construction due to Baumslag (1973,J. Austral. Math. Soc.16, 98–110), we prove that a free metabelian group of finite rank can be embedded in a finitely presented metabelian group that is also residually torsion-free nilpotent