Hyperbolic cone-manifold structures with prescribed holonomy I: punctured tori
Abstract
We consider the relationship between hyperbolic cone-manifold structures on surfaces, and algebraic representations of the fundamental group into a group of isometries. A hyperbolic cone-manifold structure on a surface, with all interior cone angles being integer multiples of $2\pi$, determines a holonomy representation of the fundamental group. We ask, conversely, when a representation of the fundamental group is the holonomy of a hyperbolic cone-manifold structure. In this paper we prove results for the punctured torus; in the sequel, for higher genus surfaces. We show that a representation of the fundamental group of a punctured torus is a holonomy representation of a hyperbolic cone-manifold structure with no interior cone points and a single corner point if and only if it is not virtually abelian. We construct a pentagonal fundamental domain for hyperbolic structures, from the geometry of a representation. Our techniques involve the universal covering group of the group of orientation-preserving isometries of the hyperbolic plane, and Markoff moves arising from the action of the mapping class group on the character variety.
- Publication:
-
arXiv e-prints
- Pub Date:
- June 2010
- DOI:
- 10.48550/arXiv.1006.5223
- arXiv:
- arXiv:1006.5223
- Bibcode:
- 2010arXiv1006.5223M
- Keywords:
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- Mathematics - Geometric Topology;
- 57M50
- E-Print:
- v.2: 41 pages, 42 figures. Improvements in graphics and exposition, incorporating referee comments. To appear in Geometriae Dedicata