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Kulkarni and Shalen [6] explained topologically Ahlfors' ίiniteness theorem and its two major supplements, the area-inequalities [2] and the finiteness of the cusps [16], considering the core of the quotient 3-manifold of finitely generated torsion-free Kleinian groups. As a consequence, they obtained the sharp inequalities on the number of cusps. In this article we treat chiefly another result from the finiteness theorem. It is that every component subgroup of a finitely generated Kleinian group is also finitely generated. This assertion is qualitative. So we will estimate quantitatively the minimal number of generators of component subgroups. Our consequence (§2) is;

Certain estimates on Kleinian groups by the core of their quotient $3$-manifold