  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 build upon previous work with
punctured tori to prove results for higher genus surfaces. Our techniques
construct fundamental domains for hyperbolic cone-manifold structures, from the
geometry of a representation. Central to these techniques are the Euler class
of a representation, the group $\widetilde{PSL_2\R}$, the twist of hyperbolic
isometries, and character varieties. We consider the action of the outer
automorphism and related groups on the character variety, which is
measure-preserving with respect to a natural measure derived from its
symplectic structure, and ergodic in certain regions. Under various hypotheses,
we almost surely or surely obtain a hyperbolic cone-manifold structure with
prescribed holonomy.
