Pleating coordinates for a slice of the deformation space of a hyperbolic 3-manifold with compressible boundary.

Abstract

Let $G\sb{\rho}$ denote the Kleinian group with presentation$$\langle T\sb1, T\sb{i}, E\sb{\rho}, E\sb{i\rho}\mid\lbrack T\sb1, T\sb{i}\rbrack=1, \lbrack E\sb{\rho}, E\sb{i\rho}\rbrack=1\rangle.$$Let $\Omega(G\sb{\rho})$ be its region of discontinuity, $\Lambda(G\sb{\rho})$ be its limit set and $C\sb{\rho}$ be the convex hull of $\Lambda(G\sb{\rho}).$ Let ${\cal S}$ be a subset of C consisting of all $\rho$ for which ${{\bf H}}\sp3\cup\Omega(G\sb{\rho})/G\sb{\rho}$ is a 3-manifold with two cusps $\Omega(G\sb{\rho})/G\sb{\rho}$ is a genus two surface. ${\cal S}$ is a two-dimensional slice of the six dimensional deformation space of $G\sb{\rho}.$ Using Bers' theorem, we show that ${\cal S}$ is topologically an open annulus. All the groups in ${\cal S}$ are normalized by $R\sb{\pi/2}, $ the rotation about the origin through an angle of $\pi/2.$ Consequently, $R\sb{\pi/2}$ preserves the bending lamination. The quotient of the genus two surface $\Omega(G\sb{\rho})/G\sb{\rho}$ by the action of $R\sb{\pi/2}, $ is an orbifold ${\cal O}$ whose underlying topological space is the 2-sphere, and which contains two order 2 cone points and two order 4 cone points. Any lamination that is invariant under $R\sb{\pi/2}$ projects to a lamination on ${\cal O}.$ We show that the set of geodesic laminations that are invariant under $R\sb{\pi/2}$ can be identified with ${\bf R}\cup\{\infty\}, $ and that all except the one corresponding to $\infty$ occur as bending laminations. Coordinates have been introduced on ${\cal S}$ that reflect the geometry of the pleated surface $\partial C\sb{\rho}/G\sb{\rho}.$ The first coordinate represents the bending lamination and the second the normalized length of the bending lamination. The set of groups corresponding to a fixed $\lambda\in{\bf R}, $ coincides with a branch of the real locus of an analytic function. For $\lambda\in{\bf Q}, $ the lamination consists of closed geodesics, and the analytic function corresponds to the traces of the group elements that represent these closed curves. All these branches are disjoint and non-singular and it can be shown that the rational branches are dense in ${\cal S}$ by interpolating them with rays along which the pleating locus is an irrational lamination.