Vol. 41, No. 2, 1972

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A general solution of binary homogeneous equations over free groups

M. J. Wicks

Vol. 41 (1972), No. 2, 543–561

Let be the free group generated by vaγiablesX,Y , 𝒞 be any given free group and W,g be elements of ,𝒞, respectively. Then x = (x,y) is a solution of the binary homogeneous equation W = g if and only if W(x) = g, where x,y are elements of 𝒞. A general method is obtained which will determine, for arbitrarily given W,g, the solution set of W = g. Effectiveness is an essential requirement for the method. It may be noted that the problem can also be regarded as the problem of finding embeddings of into 𝒞. The problem of solving an equation splits naturally into two parts: the existence problem, to determine whether there are solutions; and secondly, the characterization of the general solution. The existence problem here is solved by showing that the equation has a solution if and only if at least one of a finite set of identities has a solution. The set of identities may be obtained in an effective (and quite practical) way from W,g. In order to solve the second part of the problem, it is necessary to analyse the way in which a solution of an identity generates solutions of the equation. This is elucidated by the introduction of a set of mappings Φ(W), the members of which are (derived from) the automorphisms of . The members of Φ(W) serve as parameters for the general solution. It has not been possible to specify Φ(W) in an effective way (at least, not according to one interpretation of this requirement), but Φ(W) is a group, and this fact can be used in applications of the theory.

Mathematical Subject Classification 2000
Primary: 20E05
Received: 28 June 1971
Revised: 17 November 1971
Published: 1 May 1972
M. J. Wicks