Two parabolic generator Kleinian groups (Q1121402)

The main aim of the work under review is to prove that every torsion free Kleinian group G of the second kind generated by two parabolic transformations is geometrically finite (Theorem 1). The conclusion of this theorem fails if the assumption that G be of the second kind is dropped (Theorem 2). Normalizing G suitably it may be assumed without loss of generality that \(G=G_{\sigma}\) is generated by \[ A(z)=z+1\quad and\quad B(z)=B_{\sigma}(z)=z/(\sigma z+1). \] Then Theorem 3 says: The set of points \(\sigma\in {\mathbb{C}}\) for which \(G_{\sigma}\) is Kleinian of the second kind is path connected.