  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.
