Hyperbolic conemanifold structures with prescribed holonomy I: punctured tori
Abstract
We consider the relationship between hyperbolic conemanifold structures on surfaces, and algebraic representations of the fundamental group into a group of isometries. A hyperbolic conemanifold 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 conemanifold 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 conemanifold 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 orientationpreserving isometries of the hyperbolic plane, and Markoff moves arising from the action of the mapping class group on the character variety.
 Publication:

arXiv eprints
 Pub Date:
 June 2010
 arXiv:
 arXiv:1006.5223
 Bibcode:
 2010arXiv1006.5223M
 Keywords:

 Mathematics  Geometric Topology;
 57M50
 EPrint:
 v.2: 41 pages, 42 figures. Improvements in graphics and exposition, incorporating referee comments. To appear in Geometriae Dedicata