Normal subgroups of limit groups of prime index

TL;DR

This paper investigates the minimal number of generators for normal subgroups of prime index in non-abelian limit groups, providing affirmative answers under certain conditions and advancing understanding of their algebraic structure.

Contribution

It establishes new results on the minimal number of generators for normal subgroups of prime index in limit groups, especially relating to the rational rank and specific properties of the group.

Findings

Affirmative answer for the rational rank case.

Conditions under which the original question has an affirmative answer.

Results applicable when the abelianization is torsion free or the group has the IF-property.

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).