A family of non-injective skinning maps with critical points
Abstract
Certain classes of 3-manifolds, following Thurston, give rise to a 'skinning map', a self-map of the Teichmüller space of the boundary. This paper examines the skinning map of a 3-manifold M, a genus-2 handlebody with two rank-1 cusps. We exploit an orientation-reversing isometry of M to conclude that the skinning map associated to M sends a specified path to itself, and use estimates on extremal length functions to show non-monotonicity and the existence of a critical point. A family of finite covers of M produces examples of non-immersion skinning maps on the Teichmüller spaces of surfaces in each even genus, and with either 4 or 6 punctures.
- Publication:
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arXiv e-prints
- Pub Date:
- December 2012
- DOI:
- 10.48550/arXiv.1212.6210
- arXiv:
- arXiv:1212.6210
- Bibcode:
- 2012arXiv1212.6210G
- Keywords:
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- Mathematics - Geometric Topology
- E-Print:
- 33 pages, 16 figures, a Mathematica notebook containing some numerical computations available as an ancillary file