Title:

Slices of quasiFuchsian space

In Chapter 1 we present the background material about curves on surfaces. In particular we define the DehnThurston coordinates for the set S = S(Σ) of free homotopy class of multicurves on the surface Σ. We also prove new results, like the precise relationship between Penner's and D. Thurston's definition of the twist coordinate and the formula for calculating the Thurston's symplectic form using DehnThurston coordinates. For Chapter 2, let Σ be a surface of negative Euler characteristic together with a pants decomposition PC. Kra's plumbing construction endows Σ with a projective structure as follows. Replace each pair of pants by a triply punctured sphere and glue, or `plumb', adjacent pants by gluing punctured disk neighbourhoods of the punctures. The gluing across the ith pants curve is denied by a complex parameter μi ∈ C. The associated holonomy representation ρμ : π1 (Σ)> PSL(2;C) gives a projective structure on Σ which depends holomorphically on the μi. In particular, the traces of all elements ρ μ (γ), where γ ∈ π1 (Σ), are polynomials in the μi. Generalising results proved in [24; 40] for the once and twice punctured torus respectively, we prove in Chapter 2 a formula giving a simple linear relationship between the coefficients of the top terms of Tr ρμ (λ ), as polynomials in the μi, and the DehnThurston coordinates of relative to PC. We call this formula the Top Terms' Relationship. In Chapter 3, applying the Top Terms' Relationship, we determine the asymptotic directions of pleating rays in the Maskit embedding of a hyperbolic surface Σ as the bending measure of the `top' surface in the boundary of the convex core tends to zero. The Maskit embedding M of a surface Σ is the space of geometrically finite groups on the boundary of Quasifuchsian space for which the `top' end is homeomorphic to Σ, while the `bottom' end consists of triply punctured spheres, the remains of Σ when the pants curves have been pinched. Given a projective measured lamination [η] on Σ, the pleating ray P = P[η] is the set of groups in M for which the bending measure pl+(G) of the top component ∂C+ of the boundary of the convex core of the associated 3manifold H3=G is in the class [η].
