The Maskit embedding of the twice punctured torus

Series, Caroline¹

¹ Mathematics Institute, University of Warwick, Coventry CV4 7AL, UK

Geometry & topology, Tome 14 (2010) no. 4, pp. 1941-1991

Résumé

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 a set of pants curves have been pinched. As such representations vary in the character variety, the conformal structure on the top side varies over the Teichmüller space T (Σ).

We investigate ℳ when Σ is a twice punctured torus, using the method of pleating rays. Fix a projective measure class [μ] supported on closed curves on Σ. The pleating ray P[μ] consists of those groups in ℳ for which the bending measure of the top component of the convex hull boundary of the associated 3–manifold is in [μ]. It is known that P is a real 1–submanifold of ℳ. Our main result is a formula for the asymptotic direction of P in ℳ as the bending measure tends to zero, in terms of natural parameters for the complex 2–dimensional representation space ℛ and the Dehn–Thurston coordinates of the support curves to [μ] relative to the pinched curves on the bottom side. This leads to a method of locating ℳ in ℛ.

Détail

DOI : 10.2140/gt.2010.14.1941

Keywords: Kleinian group, Maskit embedding, bending lamination, pleating ray, representation variety

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