A quadratic parabolic group

When k is a 2-bridge knot with group π K , there are parabolic representations (p-reps) θ: π K → PSL( ): = PSL(2, ). The most obvious problem that this suggests is the determination of a presentation for an image group π K θ. We shall settle the easiest outstanding case in section 2 below, viz. k the figure-eight knot 4 1 , which has the 2-bridge normal form (5, 3). We shall prove that the (two equivalent) p -reps θ for this knot are isomorphisms of π K on π K θ. Furthermore, the universal covering space of S 3 \ k can be realized as Poincaré's upper half space 3 , and π K θ is a group of hyperbolic isometries of 3 which is also the deck transformation group of the covering 3 → S 3 \ k . The group π K θ is a subgroup of two closely related groups that we study in section 3. We shall give fundamental domains, presentations, and other information for all these groups.