Equations in free metabelian groups

Abstract

Baumslag and Mahler [1 ] have shown that, for F a free group and hence G = F/F a free metabelian group, and p any prime, the relation aPbP = cP cannot hold for elements a, b, and c of G such that aG' and bG' are independent elements of the free abelian group G/G'. In answer to a question they raised, we show by their methods that, if p, q, and r are three primes, not all the same, then there exist solutions of the equation aPb q = cr in G, with a and b independent modulo G'. We may suppose that r#p, q. If such a solution exists at all, one exists in G, free metabelian on two generators x and y, and such that, modulo G', a xmmr, b yfl, and c=xmPyn, for some positive integers m and n. Let L be the ring of Laurent polynomials over the integers in x and y (that is, admitting both positive and negative integer exponents). Then G' is naturally the free L module with generator k =x xly' ixy, that is, with ux=X lux, uy =y'luy, and UA+B =uAuB, for all u in G' and A, B in L. In this notation, we have