Final Activity Report Summary - NKHYPORB (2-generator Discrete Groups and Hyperbolic Orbifolds)
Discreteness criteria (necessary and sufficient conditions) are found for the non-elementary subgroups of PSL(2,C) without invariant plane in case when f is pi-loxodromic, g is parabolic or elliptic, and g has an invariant plane orthogonal to the axis of f. The orbifolds uniformised by such groups G are obtained, group presentations for G are found, and the trace parameters are computed.
Since discreteness criteria are known for the subgroups of PSL(2,C) with a parabolic generator f and an elliptic, parabolic, or hyperbolic generator g, whose commutator has real trace, the result obtained has enabled us to describe all real points of the parameter space of two-generator Kleinian groups with a parabolic generator, that is to describe a certain two-dimensional slice through this space.
In addition, a complete list of orbifolds uniformised by non-elementary two-generator subgroups of PSL(2,C) without invariant planes whose generators and their commutator have real traces is obtained.
Since discreteness criteria are known for the subgroups of PSL(2,C) with a parabolic generator f and an elliptic, parabolic, or hyperbolic generator g, whose commutator has real trace, the result obtained has enabled us to describe all real points of the parameter space of two-generator Kleinian groups with a parabolic generator, that is to describe a certain two-dimensional slice through this space.
In addition, a complete list of orbifolds uniformised by non-elementary two-generator subgroups of PSL(2,C) without invariant planes whose generators and their commutator have real traces is obtained.