Gluing construction and Maskit slice

In this work we mainly deal with Kleinian groups, which are discrete groups of isometries of the hyperbolic 3–space. In the 1960s, Kleinian groups were studied mainly analytically, but in the 1970s Thurston revolutionised the subject by taking a more topological viewpoint. In 1990s Keen and Series introduced the Pleating Coordinates Theory. Their key idea was to study the deformation spaces of holomorphic families of Kleinian groups via the internal geometry of the associated hyperbolic 3–manifold. In this book, given a surface of negative Euler characteristic, we endow it with a projective structure, which depends on some complex parameters, using a `plumbing' construction. In particular, the traces of the holonomy image of the curves on S are polynomials in these parameters, and we prove a formula expressing the coefficients of the top terms of these polynomials in terms of the Dehn-Thurston coordinates of the curves. If the representation is free and discrete, then the representation lies on the Maskit slice, and the formula discussed above enables us to find the asymptotic direction of the pleating rays in the Maskit slice as the bending measure tends to zero.