The Riley slice revisited

Abstract

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 holomorphic 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 compressible 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 fixed, have two connected components.

The diagonal slice of Schottky space

“…Step 1: Enumeration of curves on ∂S. We need to enumerate essential nonperipheral unoriented simple curves on ∂S up to homotopy equivalence in S. As is well known, such curves on ∂S are, up to homotopy equivalence in ∂S, in bijective correspondence with lines of rational slope in the plane, that is, with Q ∪ ∞, see for example [15,19]. For (p, q) relatively prime and q ≥ 0, denote the class corresponding to p/q by γ p/q .…”

Homotopically equivalent simple loops on 2-bridge spheres in 2-bridge link complements (I)

“…Then each J i is a connected and simply connected [4,4]-map such that M = J 1 ∪ · · · ∪ J n . Moreover σ = σ 1 ∪ · · · ∪ σ n and τ = τ 1 ∪ · · · ∪ τ n .…”

“…Next suppose that σ ∩ τ consists of a single vertex, say v 0 . Cut M open at v 0 to get a connected and simply connected [4,4]-map M ′ . In this process, the vertex v 0 is separated into two distinct vertices, say v ′ 0 and v ′′ 0 , in M ′ such that…”

“…(2) Let r = 8/35 = [4, 2, 1, 2]. Again by Lemma 3.1, we obtain that the S-sequence ofû r is S(û r ) =(4,4,5,4,4,5,4,4). By the formula for u r in Lemma 3.1, this implies S(r) = S(u r ) =(5,4…”

“…Let r = 8/35 = [4, 2, 1, 2]. Recall also from Example 3.10 that S(r) =(5,4,5,4,4,5,4,4,5,4,5,4,4,5,4,4).…”