The Maskit embedding of the twice punctured torus

Series, Caroline

doi:10.2140/gt.2010.14.1941

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The Maskit embedding of the twice punctured torus

Caroline Series

Geometry & Topology 14 (2010) 1941–1991

DOI: 10.2140/gt.2010.14.1941

Abstract

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 $ℛ$ .

Keywords

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

Mathematical Subject Classification 2000

Primary: 30F40

Secondary: 30F60, 57M50

References

Publication

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

Authors

Caroline Series
Mathematics Institute University of Warwick Coventry CV4 7AL UK
http://www.warwick.ac.uk/~masbb

Volume 14, issue 4 (2010)