(40 pages) – on the arXivpublished in Geometriae Dedicata.

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 \(\widetilde{PSL_2\mathbb{R}}\) of the group of orientation-preserving isometries of \(\mathbb{H}^2\) and Markoff moves arising from the action of the mapping class group on the character variety.

Hyperbolic cone-manifold structures with prescribed holonomy I: punctured tori

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