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The Maskit embedding of the twice punctured torus

Caroline Series

Geometry & Topology 14 (2010) 1941–1991

The Maskit embedding 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 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 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 . Our main result is a formula for the asymptotic direction of P in as the bending measure tends to zero, in terms of natural parameters for the complex 2–dimensional representation space 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 in .

Kleinian group, Maskit embedding, bending lamination, pleating ray, representation variety
Mathematical Subject Classification 2000
Primary: 30F40
Secondary: 30F60, 57M50
Received: 6 January 2009
Revised: 30 June 2010
Accepted: 3 June 2010
Published: 29 August 2010
Proposed: Jean-Pierre Otal
Seconded: Colin Rourke, Joan Birman
Caroline Series
Mathematics Institute
University of Warwick
Coventry CV4 7AL