A two-dimensional slice through the parameter space of two-generator Kleinian groups

Abstract

We describe all real points of the parameter space of two-generator Kleinian groups with a parabolic generator, that is, we describe a certain two-dimensional slice through this space. In order to do this we gather together known discreteness criteria for two-generator groups and present them in the form of conditions on parameters. We complete the description by giving discreteness criteria for groups generated by a parabolic and a $\pi$-loxodromic elements whose commutator has real trace and present all orbifolds uniformized by such groups.