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The Maskit embedding of the twice punctured torus
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Series, Caroline (2010) The Maskit embedding of the twice punctured torus. Geometry & Topology, Vol.14 (No.4). pp. 1941-1991. doi:10.2140/gt.2010.14.1941 ISSN 1364-0380.
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Official URL: http://dx.doi.org/10.2140/gt.2010.14.1941
Abstract
The Maskit embedding M of a surface Sigma is the space of geometrically finite groups on the boundary of quasifuchsian space for which the "top" end is homeomorphic to Sigma, while the "bottom" end consists of triply punctured spheres, the remains of Sigma 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 Teichmuller space T(Sigma).
We investigate M when Sigma is a twice punctured torus, using the method of pleating rays. Fix a projective measure class [mu] supported on closed curves on Sigma. The pleating ray P-[mu] 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 [mu]. 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 [mu] relative to the pinched curves on the bottom side. This leads to a method of locating M in R
Item Type: | Journal Article | ||||
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Subjects: | Q Science > QA Mathematics | ||||
Divisions: | Faculty of Science, Engineering and Medicine > Science > Mathematics | ||||
Journal or Publication Title: | Geometry & Topology | ||||
Publisher: | Geometry & Topology Publications | ||||
ISSN: | 1364-0380 | ||||
Official Date: | 2010 | ||||
Dates: |
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Volume: | Vol.14 | ||||
Number: | No.4 | ||||
Number of Pages: | 51 | ||||
Page Range: | pp. 1941-1991 | ||||
DOI: | 10.2140/gt.2010.14.1941 | ||||
Status: | Peer Reviewed | ||||
Publication Status: | Published | ||||
Access rights to Published version: | Restricted or Subscription Access |
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