Normal subgroups in limit groups of prime index

Motivated by their study of pro-p limit groups, D.H. Kochloukova and P. A. Zalesskii formulated in [15, Remark after Theorem 3.3] a question concerning the minimum number of generators d(N) of a normal subgroup N of prime index p in a non- abelian limit group G (see Question*). It is shown that the analogous question for the rational rank has an affirmative answer (see Theorem A). From this result one may conclude that the original question of Kochloukova and Zalesskii has an affirmative answer if the abelianization G^ab of G is torsion free and d(G)= G(G^ab) (see Corollary B), or if G is a special kind of one-relator group (see Theorem D)