Normal subgroups of limit groups of prime index

Abstract

Motivated by their study of pro-p limit groups, D.H. Kochloukova and P.A. Zalesskii formulated 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 (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).