I am doing some rank control about the fundamental group of a 3 dim hyperbolic orbifold. After cuting out the regular neighborhood of all the singularity, I get a manifold with imcompressible boundary. I can control the rank of this manifold from below using homology of the boundary and "half live half die theorem". But then I glue the singluarity back, this is equal to gluing a disk along n (n>1) times of simple cloesed curve on the imcompressible boundary. Will this operation reduce the rank of the fundamental group? Thanks very much!