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In Chapter 1 we present the background material about curves on surfaces. In particular\ud we define the Dehn-Thurston coordinates for the set S = S(Σ) of free homotopy class\ud of multicurves on the surface Σ. We also prove new results, like the precise relationship\ud between Penner's and D. Thurston's definition of the twist coordinate and the formula for\ud calculating the Thurston's symplectic form using Dehn-Thurston coordinates.\ud For Chapter 2, let Σ be a surface of negative Euler characteristic together with a pants\ud decomposition PC. Kra's plumbing construction endows Σ with a projective structure as\ud follows. Replace each pair of pants by a triply punctured sphere and glue, or `plumb',\ud adjacent pants by gluing punctured disk neighbourhoods of the punctures. The gluing across\ud the ith pants curve is denied by a complex parameter μi ∈ C. The associated holonomy\ud representation ρμ : π1 (Σ)--> PSL(2;C) gives a projective structure on Σ which depends\ud holomorphically on the μi. In particular, the traces of all elements ρ\ud μ (γ), where \ud γ ∈ π1 (Σ),\ud are polynomials in the μi.\ud Generalising results proved in [24; 40] for the once and twice punctured torus respectively,\ud we prove in Chapter 2 a formula giving a simple linear relationship between the coefficients\ud of the top terms of Tr ρμ (λ\ud ), as polynomials in the μi, and the Dehn-Thurston coordinates\ud of \ud relative to PC. We call this formula the Top Terms' Relationship.\ud In Chapter 3, applying the Top Terms' Relationship, we determine the asymptotic directions\ud of pleating rays in the Maskit embedding of a hyperbolic surface Σ as the bending\ud measure of the `top' surface in the boundary of the convex core tends to zero. The Maskit\ud embedding M of a surface Σ is the space of geometrically finite groups on the boundary\ud of Quasifuchsian space for which the `top' end is homeomorphic to Σ, while the `bottom'\ud end consists of triply punctured spheres, the remains of Σ when the pants curves have been\ud pinched. Given a projective measured lamination [η] on Σ, the pleating ray P = P[η] is the\ud set of groups in M for which the bending measure pl+(G) of the top component ∂C+ of the\ud boundary of the convex core of the associated 3-manifold H3=G is in the class [η].
[14] David Dumas. Complex projective structures. In Handbook of Teichmuller theory. Vol. II, volume 13 of IRMA Lect. Math. Theor. Phys., pages 455{508. Eur. Math. Soc., Zurich, 2009.
[15] D. B. A. Epstein. Transversely hyperbolic 1-dimensional foliations. Asterisque, (116):53{69, 1984. Transversal structure of foliations (Toulouse, 1982).
[38] Robert C. Penner and John L. Harer. Combinatorics of train tracks, volume 125 of Annals of Mathematics Studies. Princeton University Press, Princeton, NJ, 1992.
[39] Caroline Series. An extension of Wolpert's derivative formula. Paci c J. Math., 197(1):223{239, 2001.
[40] Caroline Series. The Maskit embedding of the twice punctured torus. Geom. Topol., 14(4):1941{1991, 2010.