Geometry & Topology Monographs the Riley Slice Revisited

In 4], Keen and Series analysed the theory of pleating coordinates in the context of the Riley slice of Schottky space R, the deformation space of a genus two handlebody generated by two parabolics. This theory aims to give a complete description of the deformation space of a holomor-phic family of Kleinian groups in terms of the bending lamination of the convex hull boundary of the associated three manifold. In this note, we review the present status of the theory and discuss more carefully than in 4] the enumeration of the possible bending laminations for R, complicated in this case by the fact that the associated three manifold has compress-ible boundary. We correct two complementary errors in 4], which arose from subtleties of the enumeration, in particular showing that, contrary to the assertion made in 4], the pleating rays, namely the loci in R in which the projective measure class of the bending lamination is xed, have two connected components. In 4], L Keen and C Series used their theory of pleating invariants to study the so called Riley slice of Schottky space. The Riley slice is shown to be foliated by pleating rays on which the geometry of the limit set has xed combinatorial properties. There are two rather subtle errors in 4], concerning the labelling and connectivity of these rays. While the errors do not substantially aaect the main results, the correct picture illustrates the interesting new phenomenon that pleating varieties may not be connected, as well as some delicate points related to the marking of the group. The ideas involved will apply to other examples, and we consider them to be of suucient interest to be worth discussing at some length. Besides explaining and correcting the errors, we take the opportunity to review the background and discuss some of the techniques used in 4] in more detail.