Motivated by their study of pro-p limit groups, D.H. Kochloukova and P.A. Zalesskii formulated in [14, Remark after Thm. 3.3] a question concerning the minimum number of generators d(N ) of a normal subgroup N of index p in a non-abelian limit group G (cf. Question*). It is shown that the analogous question for the rational rank has an affirmative answer (cf. Thm. A). From this result one may conclude that the original question of D.H. Kochloukova and P.A. Zalesskii has an affirmative answer if the abelianization G ab of G is torsion free and d(G) = d(G ab ) (cf. Cor. B), or if G has the IF-property (cf. Thm C).

Motivated by their study of pro-plimit 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 subgroupNof prime indexpin a non-abelian limit groupG(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^{\mathrm{ab}}}ofGis torsion free and{d(G)=d(G^{\mathrm{ab}})}(see Corollary B), or ifGis a special kind of one-relator group (see Theorem D).

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We study homological approximations of the profinite completion of a limit group (see Thm. A) and obtain the analogous of Bridson and Howie's Theorem for the profinite completion of a non-abelian limit group (see Thm. B).