On roots of \(a^ kb^{\ell}\) (Q1061227)

Using techniques first introduced by \textit{W. Magnus} [in J. Reine Angew. Math. 163, 141-165 (1930; JFM 56.0134.03)] some additional results are obtained on the problem of determining conditions on a given word \(R=R(x_ 1,...,x_ n)\) so that R shall have a root \(Q=Q(x_ 1,...,x_ n)\). (In a free group F on the free generators \(x_ 1,...,x_ n\), a word \(Q=Q(x_ 1,...,x_ n)\) is said to be a root of a word \(R=R(x_ 1,...,x_ n)\) if the normal subgroup of F generated by Q contains R.) As an example, \(R=a^ kb^{\ell}\) is examined. All roots of \(a^ kb^{\ell}\) are obtained for various k and \(\ell\), and these results are interpreted for the theory of one relator groups with centre.